In two dimensions, that's impossible, because there's no other direction for that new vector to point. In more than three dimensions, that's impossible, because there's no single direction for that new vector to point.
It's a hacky way to express the more general concept of orthogonality and handedness: The magnitude of the product we want is proportional to how orthogonal the two vectors are, and the sign is due to a handedness convention we've adopted.
Well, if two vectors are orthogonal to any degree at all, they span a two-dimensional surface, so why not define a product which takes two one-dimensional vectors and makes a signed two-dimensional surface out of them, with an area proportional to their lengths and how orthogonal they are and a sign due to a handedness convention? That's the wedge product in Clifford algebra.
The fun comes when you realize that these signed areas are quaternions and can be used to rotate vectors in arbitrary-dimensional spaces simply by picking vectors which span the desired plane of rotation and choosing apt multiplication rules.
The real fun comes when you realize that choosing your metric cleverly gives you one plane where rotations are hyperbolic instead of Euclidean (circular), so you can model accelerations as rotations in the space-time plane. Which is Special Relativity. Which is neat.
This lays it all out nicely with pictures and everything: http://www.av8n.com/physics/clifford-intro.htm
And here is SR done as hyperbolic rotations in a space-time plane: http://www.av8n.com/physics/spacetime-welcome.htm