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Magnetism doesn't use a cross product; it uses a wedge product[0]. It generalizes perfectly well to 2d, you just have <<xy>> instead of <<xy,xz,yz>>, just like you have <x,y> instead of <x,y,z>.

Edit: the reason 2d doesn't have a cross product is because you can no longer misinterpret a bivector <<xy,xz,yz>> as a (1-)vector <yz,xz,xy> by confusing each basis bivector with the basis vector[1] orthogonal to it.

0: https://en.wikipedia.org/wiki/Wedge_product

1: Actually one of the two (positive or negative) possible basis vectors; this is why cross products have a right-hand-rule versus left-hand-rule ambiguity.