The Special Orthogonal Group $SO(3) =$ {${A \in M_3(\mathbb{R}) : A^TA = I, det(A) = 1}$}
I have successfully shown that $SO(3)$ is a manifold, but I am having a difficult time explicitly finding a set of charts and their corresponding transition maps.
I understand that there is an entire Wikipedia page on this topic, but it's kind of overwhelming. I'm not sure where to begin and what the simplest method would be.