The infinitesimal surface vector you're referring to dA is always perpendicular to the surface element it corresponds to by definition. A similar infinitesimal volume vector dV would have to be perpendicular to the volume element somehow, which is hard to do unless you bring in four spacial dimensions. I'm not sure this will be a viable approach except perhaps in very high level math/physics. I myself do not have any real experience with 4D vectors.
