The Levi-Civita tensor is also know as the third rank fully
antisymmetric unit tensor and is defined by:
Using this we can reduce the cross product to the following tensor contraction, using the Einstein summation convention:
where (as before) we sum repeated indices over all of the orthogonal cartesian coordinate indices. Note well that it is understood that any leftover index in a contraction of this sort represents a component in a vector answer.