Question: I am seeking methods or references to numerically compute integral over the orthogonal group $O(d)$. The specific context is to compute integrals of the form: $$ \int_{O(d)} f(g) dg $$ where $f:O(d) \rightarrow \mathbb{R}$ is a square-integrable function and $dg$ represent the Haar measure on $O(d)$. Note that $f$ is quite involved in my application, and I am seeking the methods regardless of the specific forms of $f$.
Thoughts: I am considering discretizing the orthogonal group and generating a grid over it. The challenge for me lies in parameterizing the elements of $O(d)$ in such a way that allows for creating a grid to approximate the integral, especially for cases where $d>2$.
For $O(2)$, a grid can be constructed using a single angle parameter, accounting for rotations and reflections. For $O(3)$, Euler angles can be employed, though this introduces complexities due to redundancy and singularities. For general $O(d)$, the dimensionality of the problem increases significantly (and I have no idea how to parameterize it).
Any insights, references, or suggestions for further reading would be greatly appreciated.