When I wrote the above, I was under time pressure and what I wrote is misleading at best. What I should have written was: Briefly, very very briefly, it unites the scalar product and a cross product-like operation called the wedge product into one operation called the geometric product.Geometric algebra appears to have some nice features. Briefly, very very briefly, it unites the scalar product and the cross product into one operation called the geometric product.
There are significant differences between the cross product and geometric algebra's wedge product. For one, the cross product cannot be defined in spaces of more than three dimensions while the wedge product can be. Further, the wedge product allows the definition of geometric entities whose square is -1 thus allowing one to write equations without using the imaginary unit.