As mentioned elsewhere on this site, there is an elegant and easily readable complete proof of a generalized Stokes's theorem in Sauvigny, **[Partial Differential Equations][1]**, volume 1, p. 38, which applies to domains in Euclidean space, as long as the boundary has capacity zero, and the form is $C^1$ on the interior, continuous to the boundary.


  [1]: https://mathscinet-ams-org.ucc.idm.oclc.org/mathscinet-getitem?mr=2254749