This is in page 381, of John Lee's Introuduction to smooth manifold of his proof of Prop 15.5.
Prop 15.5: If $M$ is given an orientation, then there is a smooth nonvanishing $n$-form on $M$ that is positively oriented at each point.
He claimed the following is true:
Let $M$ be an oriented $n$- manifold. Then $\wedge^n_+ T^*M $ is open in $\wedge^nT^*M$ where $\wedge^n_+T^*M$ is the subset consisting of positively orineted $n$-covectors at all points of $M$.
Why is it so? I have written what I think constitute as a proof below. Feel free to criticize.
Proof:
If $M$ is orietned $n$-manifold. There exists local frame $\sigma$ on a neighborhood $U$, which is positively oriented. This corresponds to a local trivlization, $$ \Phi: \pi^{-1}(U) \rightarrow U \times \Bbb R, v_p\sigma(p) \mapsto (p,v_p). $$ Where $v_p \in \Bbb R$. Then $$ \wedge^n_+T^*M \cap \pi^{-1}(U) = \Phi^{-1}(U \times \Bbb R_{>0}). $$ which is an open set. So $\wedge^n_+T^*M$ is union of open sets.