Let be $M\in Lie(SO_n(\mathbb{R}))=\{X\in M_n(\mathbb{R}):\ tr(X)=0\}$ and $y_k=\sum_{l>k}\frac{(l-k)k!}{l!}$.
Now define $N=\sum_{k=0}^{\infty}\frac{M^k y_k}{k!}$
Will N be invertible? My intuition will say that yes, it will be do to the similarity with the exponential function, but I don't have any idea to how to demonstrate it.
PS: sorry for my bad English.