Let $A$ be an $n \times n$ nondegenerate square matrix with real-valued entries. If we interpret the rows of $A$ as points in $\mathbb{R}^n$, then $A$ defines a simplex. We'll say $v \in \mathbb{R}^n$ is a normal vector to $A$ if $v$ is the normal vector to the hyperplane on which this simplex lies.
I am looking for a function $f_A: \mathbb{R}^n \to \mathbb{R}^{n \times n}$ that maps a normal vector to a rotation of $A$ that has that normal vector. It's easy to see that many rotations of $A$ might correspond to a single normal vector; thus, many implementations of $f_A$ might be possible. Any of them will do, as long as $f_A$ is continuous.
Thanks!