I know two definitions of an orientation of a smooth n-manifold $M$:
1) A continuous pointwise orientation for $M$.
2) A continuous choice of generators for the groups $H_n(M,M-\{x\})=\mathbb{Z}$.
Why are these two definitions equivalent? In other words, why is a choice of basis of $\mathbb{R}^n$ equivalent to a choice of generator of $H_n(\mathbb{R}^n,\mathbb{R}^n-\{0\})=\mathbb{Z}$?
See comments for precise definitions.
Thanks!