distribution of the orthogonal complement to a random gaussian matrix A.

Question

I was wondering if the following statement is true: Suppose $m>n$ and the entries in $A \in \mathbb{R}^{m \times n}$ are i.i.d gaussian distributed. Then $rank(A)=n$ w.p. 1. Now consider the full rank orthogonal complement matrix $A^{\perp} \in \mathbb{R}^{(m-n) \times m}$, where $A^{\bot}A = 0$. The question which has confused me is the following: Can each entry in the orthogonal complement $A^{\perp}$ also be generated i.i.d w.r.t the gaussian distribution? Thanks all.