Let $p:E \to M$ be a smooth real vector bundle of rank $r$ over a smooth manifold $M$. Suppose that $E$ is oriented and equipped with a smooth bundle metric $h$. Let $\operatorname{Fr}_{SO}(E,h)$ denote the set of pairs $(x,u)$, where $x \in M$ and $u=(u_1,\dots,u_r)$ is a positively oriented $h_x$-[orthonormal basis](/page/Orthonormal%20Basis) of the fibre $E_x$. Then the projection

\begin{align*} \pi_{SO}: \operatorname{Fr}_{SO}(E,h) \to M, \quad (x,u) \mapsto x \end{align*}

admits a canonical smooth principal $SO(r)$-bundle structure with right action given by change of oriented orthonormal basis.