In the plane, the exterior angle of a vertex is $\pi -$ the standard ("interior") angle, which may be negative in some cases. The following is true for non-weird polygons:

The sum of the exterior angles at each vertex is a full turn ($2\pi$ radians).

I am informally calling polygons with self-crossing edges or holes as "weird" -- please do let me know what the standard terminology is. I have seen an extension to 3-dimensional polytopes of this form:

The sum of the exterior angles of a polytope is $4\pi$ radians.

In this case, the exterior angle of a vertex is $2\pi -$ (the sum of the face angles at that vertex). I have not seen a proof, but I think it is true for non-weird polytopes, and a modified version is probably true for polytopes with nonzero genus.

My question is: is there a general n-dimensional version of these properties?