[pandaman]

If you define cross product through minors, then you can expand it to any dimension, just with the different number of arguments, n-1 for R^n, 1 in R^2: [{x,y},] = {y,-x} for example. 3 in R^4, etc. Going to have the same properties as R^3 wrt to linearity and anticommutativity.

[tobinfricke]

I think you are basically reinventing the exterior / wedge product, which is indeed the generalization of the cross product.