How to generate a random matrix in a basis independent way (so that the random distribution does not change if the coordinates are rotated)?
I am especially interested in generating random rotation maps (in matrix or any other form) in basis independent way, or at least independent of swapping every two unit vectors.
I am also especially interested in generating "small" random changes (so called mutations) of matrices.
Now (see the answers below) it is clear how to generate basis-independent random matrices (both small and big) and also basis independent rotation matrices.
Now the question: how to generate small (rotating to a small angle) rotation matrices, is also answered.
Are there faster algorithms?