Let $n$ and $m$ be two integers such that $m\leq n$. Let $G_{n,m}$ be the set of all m-dimensional linear subspaces of $\mathbb{R}^n$. Assume we want to generate a subspace of dimension $m$ which is distributed uniformly over $G_{n,m}$.
Claim:
Let $A$ be $n \times m$ random matrix such that each entry distributed as $N(0,1)$ (normal distribution). Let $E = \text{span}(c_1,\dots,c_m)$ where $c_i \in \mathbb{R}^n$ be the $i$-th column of $A$. Then, $E$ is distributed uniformly over $G_{n,m}$.
[I am looking for an elementary proof of this claim. This claim is stated in Chapter 5 of Vershynin's HDP book without proof.]