I am teaching a course on manifolds, and soon I will have to prove the Stokes' theorem which, of course, involves defining oriented manifolds. There are many ways to define an oriented manifold. My favorite way is by the reduction of the structure group of the tangent bundle. But this definition and a couple of other that I know give just one orientation for the point: $GL(V)/GL^{+}(V)$ is 
${\mathbb Z}/2{\mathbb Z}$ when $\dim V \ge 1$, but when $\dim V=0$ then $GL(V)$ has only one element. Of course, it is possible to define two orientations of a point by convention. I would like to know if is there any way to define the orientation for smooth manifolds in a uniform way that would also yield two orientations of a point.



orientations for zero-dimensional manifolds
I think instead of reducing the structure group of the tangent bundle, you could consider reducing the structure group of its top exterior power, which is always a one-dimensional vector bundle, even in dimension zero. Then the set of orientations of each connected component is a torsor under $GL\_1(\mathbb{R})/GL\_1^+(\mathbb{R}) \cong \{ \pm 1 \}$.


