see the appendix of this paper for understanding Haar measure: Determinantal point processes in the plane from products of random matrices
intuition for Haar random orthogonal matrix: choose a vector randomly from the unit sphere in ${\mathbb R}^n$ (uniform distribution on the unit sphere). That's the first column. Now for the second column, choose a vector randomly from the unit sphere in the $n-1$ dimensional subspace orthogonal to the first column. Similarly for the third column, choose a vector randomly from the unit sphere in the $n-2$ dimensional subspace orthogonal to the first two columns...and so on....
