I am trying to constrain the space of matrices used for the layers of a neural network to those in (). It is proven that () is a manifold. I'm trying to find a way to smoothly traverse this manifold.
Note this is a bit different than just sampling () using something like Graham-Schmidt. The function used to construct a matrix would need to be, and ideally run in linear time rather than the cubic time needed for Graham-Schmidt. The sampling of () would not need to be uniform.
I believe I read a post here before that had a function for constructing said matrices, but I haven't been able to find it again.
If I understand your question correctly you are looking for a parametrization of $SO(n)$ that you can then perhaps also use for optimization procedures. Such a method has, for example, been discussed here. I am not an expert on this, but essentially we can use the exponential map $$\exp(A) := I + A + \frac{A^2}{2} + \dots $$ to translate an optimization problem over a compact group such as $SO(n)$ into a problem over the space of skew-symmetric matrices. The latter can be parametrized more easily (see the paper for details).