Creating an Gaussian Orthogonal Matrix

Question

I am working on a project where I need to create a special type of orthogonal matrix, such that all of the rows and all columns are orthogonal, but each entry in the matrix is drawn from a Gaussian distribution with a mean of $0$ and variance of $1$.

Is this possible, and if so what would be an easy way to do it?

Answer 1

Here is a partial answer. If your matrix is $2^n\times2^n$, we can obtain a random sample from the $n$-fold Kronecker product $$ \pmatrix{1&-1\\ 1&1}\otimes\cdots\otimes\pmatrix{1&-1\\ 1&1}\otimes\pmatrix{U&-V\\ V&U} $$ where $U$ and $V$ are two i.i.d. standard normal random variables.