Cross product of basis vectors in general relativity

Question

Say I wanted to find an expression for the differential surface trapped between two basis vectors (in 4-dimensional curved coordinates), one approach would be to take their cross product, and take the absolute value of the resulting vector. But from what I understand, the cross product cannot work in 4 dimensions. In that case, in what way can the differential surface be calculated in general relativity, a theory that deals with 4-dimensional coordinates? Could one produce a tensor from the cross product of the basis vectors (just as the dot product of the basis vectors produces the metric tensor)? would it have any significant physical meaning?

Answer 1

The parallelogram spanned by two vectors - and more generally the generalized volume spanned by $n$ vectors - is expressed by the wedge product of these vectors.

The cross product in three dimensions is the concatenation of the wedge product and the Hodge dual.