Is There a Standard Parametrization of the Defining Representation of $\mathrm{SO}(N)$?

Question

Is there a standard parametrization for the space of rotations of $N$-dimensional vectors, the defining representation of $\mathrm{SO}(N)$? I know that in $3$ dimensions the two "standard" parametrizations of rotations are the Euler and Tait-Bryan angles, for instance, and I'm curious whether there is a known parametrization for $N$ dimensions that is considered standard to a similar degree. I've already cooked up a parametrization of $\mathrm{SO}(N)$ that is Euler like, including identifications of which angles are polar (canonical range $[0,\pi]$) and which are azimuthal (canonical range $[0, 2\pi)$), and fomulae for translating an arbitrary set of angles into the canonical range. I mainly want to know if it's worth the effort of publishing.

Answer 1

It may not be known as a standard, but this post about the Haar measure of $\operatorname{SO}(N)$ describes a parameterization that is identical to the one I implemented, but with a different choice of some conventions.

There is also the group theory based parameterization. It is more standard for theoretical analytic work than numerical work, but it is explicitly computable. Let $M_{ij} = -M_{ji}$ be a skew-symmetric matrix. Calculating the operator exponential of $M$ (a built in function in Julia and MATLAB) yields a rotation matrix.