Given a diffeomorphism $F:M\times\mathbb{R}^n\longrightarrow TM$ such that $F(\{p\}\times\mathbb{R}^n)=T_pM$ for all $p\in M$ and for every $p\in M$ the map $v\in\mathbb{R}^n\mapsto F(p,v)\in T_pM$ is a diffeomorphism, do we have $n$ vector fields $X_1,\dots,X_n$ on $M$ such that $X_1(p),\dots,X_n(p)$ span $T_pM$ for all $p\in M$?
I know this is true for $v\in\mathbb{R}^n\mapsto F(p,v)\in T_pM$ being a linear isomorphism and feel that it should also be true for compact $M$ (maybe not even this?). But for more general nonlinear diffeomorphism $F$ and non-compact $M$, is this statement still true?