Let $G$ be a Lie group with identity element $e$ and dimension $n$. I know we can take a non-vanishing $n$-form (call it $\omega$) on the vector space $T_eG$. For a Lie group, as we have an isomorphism between $Lie(G)$ (the set of left invariant vector fields on $G$) and $T_eG$, we can define an $n$ form $\chi$ on $G$ by the following process. Let $L_{g^{-1}}$ be the function denoting left multiplication of an element by $g^{-1}$. Then define:
$$\chi(g)=(L_{g^{-1}})^\ast \omega$$
and this is a non-vanishing $n$-form on $G$ as the map $L_{g^{-1}}$ is a diffeomorphism so takes non-vanishing forms to non-vanishing forms.
My question is can we extend this idea to all parallelizable manifolds? I understand from having read it somewhere that all parallelizable manifolds are orientable, so would this sort of idea work in the more general case, or do we have to use a different method?
Thanks