I need a reference for a definition of an integral of a function $f:\partial\Omega \to \mathbb{R}$ over the boundary of a Lipschitz open domain $\Omega \subset \mathbb{R}^n$ (the usual domain in PDEs): $$\int_{\partial\Omega}fd\sigma = ?$$
I am aware of two definitions: one in Necas' book (which is painful to read) and one in Boyer's book which uses a lot of half-space stuff which I want to avoid.
I would like a definition that uses chart maps back to $\mathbb{R}^{n-1}$ obviously. I have searched a long time but haven't found anything good to cite.
Thanks for any help.