Generalization of rectifiable curves and integrals over them to higher dimensions

Question

Length of a curve can be defined for an arbitrary rectifiable curve(even in an arbitrary metric space). As is shown in this answer we can define line integral over any such curve(even in an arbitrary metric space). When it comes to surfaces and higher-dimensional cases it seems that there are no constructions of such generality. Wherever "surface area" and "surface integrals" are defined, some kind of smoothness is required for the surface in question. Are there any definitions of "n-dimensional surface area" and "n-dimensional surface integral"(which resemble the definitions of length and line integral over an arbitrary rectifiable curve) with much weaker conditions on the "surface" (in the same way as "being rectifiable" is weaker than "being smooth" for a curve) ?