Let $\pi:E\to M$ be a smooth real vector bundle of rank $r\in\mathbb N$ over a smooth manifold $M$. Let $(U_i)_{i\in I}$ be an open cover of $M$ with smooth local trivializations
More precisely, after identifying $\Lambda^r\mathbb R^r$ with $\mathbb R$ by sending $e_1\wedge\cdots\wedge e_r$ to $1$, the transition function of $\Lambda^r E$ from the trivialization induced by $\varphi_j$ to the trivialization induced by $\varphi_i$ is the scalar map $x\mapsto \det(g_{ij}(x))$.