I am trying to gain an intuitive understanding of what these classes represent (for $n > 2$). What do the elements of the same conjugacy class have in common? Can they be related by some notion of "amount of rotation"?
For example, if an element $g \in SO(n)$ rotates all vectors in $S^{n-1}$ by at most some angle $\theta$, then surely the elements in the conjugacy class must have some similar property, no?