Let $M$ be a smooth $n$-manifold, and let $\operatorname{Fr}(TM)$ denote the frame bundle of $TM$, whose fibre over $x \in M$ consists of linear isomorphisms $u: \mathbb{R}^n \to T_xM$. There is a natural bijection between nowhere-vanishing smooth $n$-forms $\omega \in \Omega^n(M)$ and principal $SL_n(\mathbb{R})$-reductions $Q \subset \operatorname{Fr}(TM)$.

Under this bijection, a nowhere-vanishing smooth $n$-form $\omega$ is sent to the subbundle

\begin{align*} Q_\omega = \{u \in \operatorname{Fr}(TM) : \omega_{\pi(u)}(u(e_1), \dots, u(e_n)) = 1\}, \end{align*}

where $\pi: \operatorname{Fr}(TM) \to M$ is the bundle projection and $(e_1,\dots,e_n)$ is the standard basis of $\mathbb{R}^n$. The associated orientation is the orientation for which a frame $u: \mathbb{R}^n \to T_xM$ is positive precisely when

\begin{align*} \omega_x(u(e_1), \dots, u(e_n)) > 0. \end{align*}