Maybe not the answer you are looking for.
To get a more specific answer, more information about the function $f_i$ are
required. In some sense the question is as hard as finding an explicit form for an implicit equation.
Theoretical local state-space form
If you consider the function
$$
g_i(y_i,x_1,x_2) := y_i - f_i(y_i,x_1,x_2)
$$
and let $\widetilde p = (\widetilde y_1, \widetilde x_1, \widetilde x_2)$ be a point with $g(\widetilde p) = 0$.
Then, an application of the implicit function theorem yields a function $\phi_i(x_1,x_2)$ locally defined around $\widetilde p$, such that for all $(y_i,x_1,x_2)$ close to $\widetilde p$ we have
$$
y_i = \phi_i(x_1,x_2) \iff y_i - f_i(y_i,x_1,x_2) = 0.
$$
Therefore, a local state space form is given by $\dot x_i = \phi_i(x_1,x_2)$.
The requirements on $f_i$ are, that $\frac{\partial f_i}{\partial y_i}(\widetilde p) \neq 0$ and that $f_i$ is (at least) continuously differentiable.
Calculus
The derivatives of $\phi_i$ can be expressed in terms of the derivatives of $f_i$,
therefore this approach also allows you to carry out some local analysis.
Theoretical global state-space form
This is in general not possible. Take
$$f(\dot x,x) = (\dot x-1)(\dot x+1),$$
obviously, the local state-space forms are either
$$
\dot x = 1 \quad \text{or} \quad \dot x = -1.
$$
But none of these are global and it depends on the initial conditions which
track you follow. The situation can be arbitrary complex.
Therefore, I would doubt that a general trick exists, which transforms you system
into an explicit equation without strict assumptions on $f_i$.
Numerics
Depending on our task, it might be sufficient to use some root-finding algorithm to get the value of $\phi_i$. (But you need to provide initial guesses for $\dot x_i$, for example the previously computed value of $\phi_i$.)
But, I'm not too familiar with numerics for implicit ODEs.
Best Answer
It is in state space form. What you are looking for is the "matrix form" and this is not possible since the system is non-linear.
Recall in the state space form, the left hand side should only have derivatives of the state, and the right hand side should have only constants, and functions of time (including external inputs)