I know that the exponential matrix of a $2\times 2$ matrix ($M$) by regarding the Euler relation is:
$$e^{-iM\hat{n}.\sigma}=\cos(M)\sigma_0+i\sin(M)\hat{n}.\sigma$$
where $\sigma=(\sigma_x.\sigma_y,\sigma_z)$ stand for the well-known $2\times 2$ Pauli matrices and $\sigma_0$ is identity matrix of size $2$.
So my question is, What would we say about a $3\times 3$ matrix? is there such a simple way to find the exponential matrix of a $3\times 3$ matrix?
For the subclass of skew-symmetric matrices there is a similar formula (Rodrigues' rotation formula), generally not. However, there are certain simplications for nilpotent and idempotent matrices. The usual way makes either use of the Cayley–Hamilton theorem or diagonalization.