The defining relation for the Clifford algebra, $Cl(1,d)$ is
$$
\{\gamma_\mu,\gamma_\nu\}=2 \eta_{\mu\nu}\ \mathbf{1}\ ,
$$
For simplicity, I will assume that $\eta_{\mu\nu}=\text{Diag}(1,-1,\ldots,-1)$ with $\mu,\nu=0,1,\ldots,d$. Other signatures can easily be incorporated. It is easy to see that $\gamma_0^2=-\gamma_i^2=\mathbf{1}$ for $i=1,\ldots,d$. Using the defining relation, one has
$$
\gamma_0 \gamma_i + \gamma_i \gamma_0 =0 \ .
$$
Multiply the above equation by $\gamma_0$ and then take the trace to obtain
$$
\text{Tr}(\gamma_i) + \text{Tr}(\gamma_0 \gamma_i \gamma_0)=0\implies \text{Tr}(\gamma_i)=0\ ,
$$
on using the cyclic property of the trace. Similarly, one can show $\text{Tr}(\gamma_0)=0$. So the defining property proves the tracelessness of the Dirac matrices.
Two representations, $\gamma_\mu$ and $\gamma_\mu'$, of the Clifford algebra are said to be equivalent if $\gamma_\mu' = S \cdot \gamma_\mu S^{-1}$ for some invertible matrix $S$.
Appendix A of the Physics Reports article by Sohnius might be a good starting point for the other properties.
Okay, this wasn't as hard to find an answer to as I expected. However, any clarifications/ critisisms are welcome.
Weinberg basically gives the answer in section 5.6 of his QFT book:
A general tensor of rank N transforms as the direct product of N (1/2, 1/2)
four-vector representations. It can therefore be decomposed (by suitable
symmetrizations and antisymmetrizations and extracting traces) into irreducible terms (A,B) with A = N/2, N/2-1,... and В = N/2, N/2-1,... . In this way, we can construct any irreducible representation (A,B) for
which A + В is an integer. The spin representations, for which A + В is
half an odd integer, can similarly be constructed from the direct product
of these tensor representations and the Dirac representation $(1/2,0)\oplus(0,1/2)$.
So it does not appear that there is any simple way to unify the representations for spinor and vector generators, but one can construct the generator for arbitrary half-spins: it has as many copies of $M$ as needed, plus one copy of $S$ if it is a half-integer.
However, the converse is not true. That is, a spin-two field must transform with two copies of $M$, but doing so does not guarantee that an object is spin-two. A counterexample is the electromagnetic tensor $F$, which is certainly spin-one. The difference lies in the ability to equate the two generators as a result of the symmetry properties of the tensor, as elaborated in this answer.
Applying this to the spin 3-2 field, we expect it to have a rotation generator that looks schematically like $S \otimes I+ I \otimes M$. And indeed this is the case- the equivalent of the Dirac equation for spin 3/2 is the Rarita-Schwinger equations:
$\gamma_a \psi^{a}_\mu=0$,
$(i \gamma^\rho \partial_\rho-m)\psi^{a}_\mu=0 $
Which transforms as $\psi'^b _\nu=(\Lambda_\nu^\mu \otimes T^b_a) \psi_\mu^a$,
whose generators are the above combination of $M$ and $S$.
Best Answer
One option is to start out with the matrix representation for two sets of conjugate Grassmann numbers (see previous thread), $\theta_i, \pi_i$ with $i=1,...,N$, such that
$\{\theta_i,\theta_j\}=0,\quad\{\pi_{i},\pi_{j}\} = 0, \quad \{\theta_i,\pi_j\} = \delta_{ij}$
Then a $2N$-dimensional Clifford algebra can be built by
$\gamma_{i}=\theta_{i}+\pi_{i}\\ \gamma_{N+i}=i(\theta_{i}-\pi_{i})$
Given the above anti-commutation relations it is straightforward to verify that $\{\gamma_{i},\gamma_{j}\}=2\delta_{ij}\mathbf{1}$. For a odd number of dimensions the last $\gamma$-matrix can be found by considering the product
$\gamma_{2N+1} = i^N\prod_{i=1}^{2N}\gamma_{i} = i^N\gamma_{1}\gamma_2...\gamma_{2N}$
To get a representation of the Dirac algebra $\{\gamma_{\mu},\gamma_\nu\}=2g_{\mu\nu}\mathbf{1}$ with signature (+,-,-,...,-) simply rotate all but one of the matrices in the representation such that $\gamma_i\to i\gamma_i$ (and relabel a bit).
This approach enables one to derive general representations of the gamma matrices from Grassmann numbers.
However, another option exists, namely to start out with a lower dimensional representation of the Clifford algebra (which can be computed by the method described above). A well-known case of a lower-dimensional representation, which was also known to Weyl & Dirac, would be the Pauli matrices:
$\sigma_{1} = \left[\begin{matrix} 0 & 1 \\ 1 & 0 \end{matrix}\right], \quad \sigma_{2} = \left[\begin{matrix} 0 & -i \\ i & 0 \end{matrix}\right], \quad \sigma_{3} = \left[\begin{matrix} 1 & 0 \\ 0 & -1 \end{matrix}\right] $
From these matrices outer products, $\rho_i = \mathbf{1}\otimes \sigma_i$ and $\eta_i = \sigma_i \otimes \mathbf{1}$, can be formed. It is then clear that $[\rho_i,\eta_j]=0$ which makes it possible to choose five matrices from the set $\{\rho_i,\eta_j,\rho_i\eta_j\}$ which fulfill the Clifford algebra.
To make this approach a bit more explicit, consider starting with a diagonal matrix from the initial set for simplicity - let us choose $\rho_3$ ($\eta_3$ would have been another option). This leaves us with two potential sets of matrices, namely $\{\rho_1,\rho_2\eta_1,\rho_2\eta_2,\rho_2\eta_3\}$ and $\{\rho_{2},\rho_{1}\eta_{1},\rho_{1}\eta_{2},\rho_{1}\eta_{3}\}$. Since $\rho_1$ is real I choose the first set, making the matrices:
\begin{align} \gamma_0 &= \left[\begin{matrix} 1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & -1 & 0 \\ 0 & 0 & 0 & -1 \end{matrix}\right] = \rho_3, &&\gamma_1 = \left[\begin{matrix} 0 & 0 & 0 & 1 \\ 0 & 0 & 1 & 0 \\ 0 & -1 & 0 & 0 \\ -1 & 0 & 0 & 0 \end{matrix}\right] = i\rho_2\eta_1 \\ \gamma_2 &= \left[\begin{matrix} 0 & 0 & 0 & -i \\ 0 & 0 & i & 0 \\ 0 & i & 0 & 0 \\ -i & 0 & 0 & 0 \end{matrix}\right] = i\rho_2\eta_2, &&\gamma_3 = \left[\begin{matrix} 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & -1 \\ -1 & 0 & 0 & 0 \\ 0 & -1 & 0 & 0 \end{matrix}\right] = i\rho_2\eta_3\\ \gamma_5 &= \left[\begin{matrix} 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \\ 1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \end{matrix}\right] = \rho_1 \end{align}
where I have taken the liberty to rotate three of them as described above. In this way the Dirac representation is found. Notice that a few choices were made along the way but that several of them can be motivated by the search for a simple representation (choosing diagonal and/or real when possible).
This approach can naturally be generalized to generate higher dimensional representations as well.