I would like to know if there is a nice way to show that, given $\vec{n}, \vec{m} \in \mathbb{R}^3$, non-parallel, any rotation $A \in SO(3)$ can be expressed as a composition of multiple rotations around $\vec{n}$ or $\vec{m}$.
I know how to show it with direct calculus for orthogonal vectors, using only a composition of 3 rotations, but I also know that the situation is more complex for non-orthogonal ones, with possibly a large number of rotations to use.