Why is the product of two vectors a vector in another dimension, why is it defined by the right-hand rule, and why is it defined by the area of the parallelogram defined by the two vectors? What are some applications of the cross product?
Let go of your visual representation. it will not take you far.
It simply is defined by the right-hand rule. Define by the left-hand rule and do everything updise down and backwards. It's a convention.
It isn't defined as an area. This is an application.