> The cross product only exists in three dimensions While true, the Wedge product is a useful concept that generalizes a cross product to arbitrary dimensions

> The cross product only exists in three dimensions While true, the Wedge product is a useful concept that generalizes a cross product to arbitrary dimensions and is used in multivariable calculus for proving various integral theorems in high dimension. Here, generalized Stokes theorems apply despite the cross product not being defined. Admittedly, it isn't really a map on the vector space, but the fact that Stokes theorems still hold makes it pretty darn useful to me.