Recently, we got introduced to the Stiefel manifold $V_k(\mathbb R^n)$, that is, the set of orthonormal $k$-frames in $\mathbb R^n$. In particular, we discussed the case for $k=2$, $V_2(\mathbb R^n) \subset \mathbb R^{2n}$. Thus, any $2$-frame consists of two $n$-dimensional orthonormal vectors $x,y\in \mathbb R^{n}$. We obtain the tree equations
\begin{align} \Vert x\Vert^2 &= 1 \\ \Vert y\Vert^2 &= 1 \\ \langle x,y\rangle &= 0 \end{align}
where $\langle \cdot,\cdot \rangle$ denotes the standard scalar product. However, we then said that we should expect $2n-3$ free parameters $$\underbrace{(n-1)}_{\text{param. for}\ x}+ \underbrace{(n-2)}_{\text{param. for}\ y} = 2n-3$$
But I don't understand what exactly is meant here, when we say $2n-3$ parameters. How can one (geometrically and algebraically) verify this? I'm assuming that this somehow relates to the $n\times k$ (here $n\times 2)$ matrices of rank $2$, but how exactly?
Sorry for the very vague question, unfortunately, we quickly skimmed over this topic and this is all we've written down. I also tried to look it up in all of my books but I didn't finy anything other than the definition of Stiefel manifolds in Kosinski's Differential Manifolds.