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, 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.

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, 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.