Quote by Magos
I'm not even sure such a thing exists :P.
Anyway, if it does, is it correct that it is:
(rotation 90 deg counter clockwise)
Couldn't find anything about it on google...

The twodimensional equivalent of a cross product is a scalar:
[tex]\hat{x} \times \hat{y} = x_{1}y_{2}x_{2}y_{1}[/tex]
It's also the determinant of the 2x2 row matrix formed by the vectors. I don't think it's usually used, though. Unlike dot products, cross products aren't geometrically generalizable to n dimensions.