In the situation of N=1, the rotation can be parameterized as $R(\theta)=e^{i\theta}$, where $\theta \in \mathbb{R}$. However, due to the periodic nature of rotation, we can constrain $\theta$ to sit in a small inverval like $(-\pi,\pi]$.
However, I have no idea about how to find a general constrain for arbitary N. Just like one dimentional rotation, SO(N) representations can be parameterized as: $$ R = e^{A} $$ Where A is an arbitary skew-hermite matrix, is it posible to find a small interval for all matrix elements so that all rotations can be represented with a limited numerical range?