The rotation group is isomorphic to the orthogonal group $SO(3)$. So a rotation matrix can represent all the possible rotation transformations on the euclidean space $R3$ obtainable by the operation of composition.
The axis/angle notation describes any rotation that can be obtained by rotating a solid object around an axis passing on the reference origin by the given angle.
Given a pair $(\mathbf{u}, \theta)$, where $\mathbf{u}$ is a vector and corresponds to the axis rotation and $\theta$ is the roation angle, can this single pair represents all the possible rotations on the euclidean space R3 obtainable by composition?
Best Answer
The answer is yes, since the exponential map $\exp: \mathbf{so}(3) \rightarrow \mathbf{SO}(3)$ is surjective (=onto).
Long answer:
Axis-angle can be represented using a $3$-vector $\omega$ while the magnitude $\theta=|\omega|$ is the rotation angle and $\mathbf{u}=^\omega/_\theta$ is the rotation axis. 3-Vectors are closed under the cross product:
$$\omega_1\in \mathbb{R}^3, \omega_2\in \mathbb{R}^3\Rightarrow (\omega_1\times \omega_2)\in\mathbb{R}^3.$$
Each such vector $\omega$ has an equivalent $3\times 3$ matrix representation $\hat{\omega}$ (which is uniquely defined by $\hat{\omega}\cdot \mathbf{a} := \omega\times \mathbf{a}$ for $\mathbf{a}$ being a general 3-vector). The space of matrices of the form $\hat{\omega}$ is called the Lie algebra $\mathbf{so}(3)$. Thus, one can show that matrices of the form $\hat{\omega}$ are closed under the Lie bracket $[A,B]=AB-BA$:
$$\hat{\omega_1}\in \mathbf{so}(3), \hat{\omega_2}\in \mathbf{so}(3)\Rightarrow [\hat{\omega}_1, \hat{\omega}_2]\in\mathbf{so}(3).$$
Now, let us consider the matrix exponential: $\exp(\mathtt{A}) = \sum_{i=0}^\infty \frac{\mathtt{A}^i}{i!} $. Two poperties can be shown:
(1) If $\hat{\omega}\in\mathbf{so}(3)$, then $\exp(\hat{\omega})\in\mathbf{SO}(3)$.
$\mathbf{SO}(3)$ is the special orthogonal group in three dimensions. Thus, it consists of matrices which are orthogonal ($\mathtt{R}\cdot \mathtt{R}^\top=\mathtt{I}$) and the determinant is 1. In other word, it is the group of pure rotations.
(2) The exponential map $\exp: \mathbf{so}(3) \rightarrow \mathbf{SO}(3)$ is surjective.
So, (1) says that every $\exp(\hat{\omega})$ is a rotation matrix. And, (2) says that for each rotation matrix $\mathtt{R}$, there is at least one axis-angle representation $\omega$ so that $\exp(\hat{\omega})=\mathtt{R}$
Proofs of (1) and (2) are in corresponding text books, e.g. [Gallier, page 24].