The rotations of $\mathbb{R}^n$ can naturally be endowed with a bi-invariant metric.
First question: is this metric the same as the one coming from the Euclidean structure of $\mathcal{M}_n(\mathbb{R})$?
Second question: for this metric, what is the injectivity radius? (Is there a chance that the injectivity radius is exactly the diameter of the manifold)
Third question: do we know how to describe a domain in anti-symmetric matrices on which the exponential is one to one?
Thanks for your help.