For cross product in 2-D, it represents the area of parallelogram. And it has an order that i on the right of the j.
Actually, cross product is in 3-D, and the result of the cross product is a vector with length of the area of parallelogram, perpendicular to the parallelogram obeying right hand rule.
Cross product can be calculated as follow.
By using duality, exist a vector p dot product [x, y, z] equals determinant.
3-D determinant means the volumn of the cube, we need to calculate component of [x, y, z] perpendicular to
w times area of parallelogram.
This equals to the vector p with the length of area of parallelogram by using duality.