Wedge Product/Cross Product?

Although they are both antisymmetric in their arguments,
the wedge product of two vectors is a bivector (a 2-index tensor);
the cross product of two vectors is another [psuedo] vector.

 

Pretty much it's just down to how you view these things.

x/\y is always defined for all x,y in any vector space, they just live in the space /\^2(V). It so happens that in the case when dim(V)=3, then /\^2(V) is (non-canonically) isomorphic to V, so people identify them, and call the resulting thing the cross product.