Can the cross product concept be completely replaced by the exterior product?

[SVN]

Do we really need concept of cross product at all? I always believed cross product to be sort of simplification of exterior product concept tailored for the 3D case. However, recently I encountered the following sentence «...but, unlike the cross product, the exterior product is associative». (https://en.wikipedia.org/wiki/Exterior_algebra) and it occurred to me that I might have been missing something.

Two remarks:
1) Clearly cross product is tied to 3D world and exterior product works for space of any dimension. Suppose we are considering 3D case only.
2) The cross product can be convenient tool. Being such it has every right to exist. But I am asking about whether it could be dropped entirely in favour of its exterior counterpart without losing anything in principle.

 

[wrobel]

Yes, cross product concept can be replaced with exterior product provided one does it correctly. But it would hardly simplify physics formulas
https://en.m.wikipedia.org/wiki/Hodge_star_operator

 

[SVN]

Thank you!

 

[mathwonk]

if you write down the definition of the exterior product of two vectors in R^3, you will see that it has the same coordinates as their cross product. However if e1,e2,e3 is the standard basis of R^3, the cross product will be expanded in terms of that same basis, while the exterior product will be expanded in the basis e2^e3, e3^e1, e2^e3. The point is that both are aspects of the determinant, in particular both products are "alternating" and multilinear, and the only determinant is essentially the only such product.

 

[atyy]

Wikipedia gives the cross product in terms of the wedge product and the hodge star:
https://en.wikipedia.org/wiki/Hodge_star_operator#Examples

 

[FactChecker]

Yes, cross product concept can be replaced with exterior product provided one does it correctly. But it would hardly simplify physics formulas
https://en.m.wikipedia.org/wiki/Hodge_star_operator

The exterior product is preferred in Geometric Algebra. Although there is a significant learning curve, GA is considered by its advocates to offer great simplifications.