Extending to a local frame that agrees with given orientation Suppose that $(e_1, \ldots, e_k)$ is an oriented basis for $T_pM$ where $M$ is an oriented Riemannian manifold. In general, we know that we can extend to a smooth local frame $(X_1, \ldots, X_k)$ on $U\ni p$ such that $X_i\rvert_p = e_i$ for each $i$. But can we further stipulate that $(X_1\rvert_q, \ldots, X_k\rvert_q)$ is oriented for each $q\in U$? I have tried thinking of ways to shrink $U$ in a suitable way. I have played around with using the orientation form $\omega$ on $M$ and with first applying Gram-Schmidt to $(X_1, \ldots, X_k)$, but no progress. Any help is much appreciated.

The local frame $(X_i)$ will agree with the chosen orientation form: if $\omega(e_1,\dots,e_k)>0$, then $\omega(X_1,\dots,X_k)>0$ in $U$, because $\omega(X_1,\dots,X_k)|_q=0$ would imply that the $X_i's$ do not form a basis of $T_qM$.