Rodrigues' rotation formula gives a convenient way to write the general rotation matrix in $R^3$

If $[v_1, v_2, v_3]$ is a unit vector on the rotation axis, and $\theta$ is the rotation angle about that axis, then the rotation matrix is given by $$ I + \sin(\theta) A +(1-\cos(\theta))A^2 $$

where $I$ is the identity matrix and $$ A = \begin{pmatrix} 0 & -v_3 & v_2 \\ v_3 & 0 & -v_1 \\ -v_2 & v_1 & 0 \end{pmatrix} $$ .