Proving the Stiefel Manifold is a Manifold

Question

My goal is to prove that the Stiefel manifold, $S(2,n) = \{(v,w) \in \mathbb{R}^n \times \mathbb{R}^n = \mathbb{R}^{2n}: |v|=|w|=1, v \perp w\}$ is a manifold by defining a diffeomorphism from $S(2,n)$ to an open subset of $\mathbb{R}^{2n-3}$. My reasoning goes like this: take a ball of radius $r$ centered at $v$ in $\mathbb{R}^n$ and intersect it with $S^{n-1}$, the unit $n-1$ sphere in $\mathbb{R}^n$. We'll write that as $U = B_r(v) \cap S^{n-1}$. Do the same thing around $w$, so $V = B_r(w) \cap S^{n-1}$. I expect $U \times V$ is an open neighborhood of the point $(v,w) \in S(2,n)$ when the radius $r$ is small enough. There exists diffeomorphisms $\phi_1$ and $\phi_2$ from $U$ and $V$, respectively, to open sets of $\mathbb{R}^{n-1}$ since $S^{n-1}$ is an $(n-1)$-dimensional smooth manifold. Thus define $\psi: U \times V \to \mathbb{R}^{n-1}\times \mathbb{R}^{n-1}$ to be $\psi(v,w) = (\phi_1(v),\phi_2(w))$. It seems natural to me to conclude that the Stiefel manifold is a $(2n-2)$-dimensional manifold, but this is wrong. What's wrong with my reasoning? What is an appropriate way to define a diffeomorphism from $S(2,n)$ to an open subset of $\mathbb{R}^{2n-3}$?