Cross Product Component Values

Question

When taking the cross product, the x component of the perpendicular vector is the (signed) area of the yz projection of the parallelogram spanned by the two vectors it's orthogonal to-right? And similarly for the other components, with the variables permuted accordingly. Why is this? Is there an intuitive reason why a perpendicular vector should be of this structure? I've heard about something like the Pythagorean Theorem with the areas, but I also don't understand this. I just think it's peculiar that the ratio of the components of a perpendicular vector are those of the two vectors' parallelogram projected onto the zy, yx, and xz planes.