@FanZheng Thank you! Could you come back and write down any other titles concerning my problem if they come to mind?

@FanZheng Could you please recommend a few books, papers concerning geometric measure theory which might be helpful in my case?

Federer's Geometric Measure Theory? There is in chapter four: Homological Integration Theory a subsection 4.5 called Normal currents of dimension $n$ in $\mathbb{R}^n$ in which Gauss-Green theorem is discussed. Is that what you mean?