It's more of an open-ended question about rotations, and I am not sure how to formulate it.
Let's say we are in a n-dimensional space (n can be large, like 20) and we compose simple rotations within that space. By simple I mean that each is in a single hyperplane 2D plane. The planes may not be orthogonal, but they all go through the origin.
Is there a simple way to characterize the rotations? Maybe re-decompose along the axes? Maybe this article applies? https://en.wikipedia.org/wiki/Orthogonal_group#Canonical_form