Plane Rotations Preserve the Dot Product

Let $n$ be a natural number with $n\ge2$ and let $g$ be a plane rotation of the Euclidean space $\mathbb{R}^{n}$. Then for all $u,v\in\mathbb{R}^{n}$, with the dot product,

Consequently, every finite composition of plane rotations $h$ satisfies $h(v)\cdot h(u)=v\cdot u$ for all $u,v$, and $h$ maps every orthonormal family to an orthonormal family.