Let $M$ be a smooth $n$-manifold. If an orientation on $M$ is given, then we can find an open cover $\{U_\alpha\}$ of $M$, and transition functions $g_{\alpha\beta}:U_{\alpha}\cap U_\beta \to \text{SO}(n)$ satisfying the cocycle condition. Suppose we have reversed the orientation on $M$. Then what is the effect on the transition functions?
For $x\in U_\alpha \cap U_\beta$, suppose $\{v_1,\dots,v_n\}$ and $\{w_1,\dots,w_n\}$ are bases for $T_xM$ corresponding to $U_\alpha$ and $U_\beta$, respectively, so that $g_{\alpha\beta}(x) (w_1,\dots,w_n)^t = (v_1,\dots,v_n)^t$. If we choose $\{-v_1,v_2,\dots,v_n\}$ and $\{-w_1,w_2,\dots,w_n\}$ as bases for the reversed orientation, then the new transition function $g_{\alpha\beta}'$ will satisfy $g_{\alpha\beta}'(x) (-w_1,w_2,\dots,w_n)^t = (-v_1,v_2,\dots,v_n)^t$, but I cannot find a relation between $g_{\alpha\beta}$ and $g_{\alpha\beta}'$.