If by "cross product" you mean "bi-linear anti-commutative product with non-trivial result in the space of vectors" it only exists in three and seven dimensions. The seven dimensional version is related to the mathematics of octions. Trivial versions can be defined in 1 and 0 dimensions. It can be proved they don't exist in other dimensions.

In other dimensions there are related products but their results are not vectors. E.g. in 2D you can define a bi-linear anti-commutative product with a 1D scalar result. It gives the area of the parallelogram with the two vectors for sides. In 4D a similar calculation yields a 6D result. An example is Plucker coordinates, where the product is calculated on 4D homogeneous coordinates.