If $M$ is a smooth, parallelizable manifold, then it is orientable.
From the definition I know, $M$ is said to be parallelizable when there is a diffeomorphism $\phi:TM\to M\times\mathbb{R}^n$ which is linear on the fibers.
What I'm trying to do is to take an orientation of $TM$ and somehow induce an orientation on $M\times\mathbb{R}^n$, which somehow would induce an orientation on $M$.
But I don't know if we always have an orientation on $TM$ and even if there is, I'm having trouble formalizing the iduced orientation.
Any tips? Thanks!