Orientation on manifold in terms of homology

Question

One can define the orientation of a manifold $M$ in terms of relative homology groups $H_n (M, M-\{p\})$ by noting that $H_n (M, M-\{p\}) \cong \mathbb Z$ and then designating a generator. Is it possible to define orientation in terms of normal homology $H_n(M)$ somehow? As far as I can tell the answer should be yes but since I could not find the orientation defined in terms of $H_n(M)$ anywhere maybe it is not possible after all.

Answer 1

This is pretty obviously impossible in general: in $\Bbb{R}^n$, for example, the top homology is trivial, so there's no way we can find two things in it to be the orientations. It works fine for closed connected manifolds though.