We can define cross products mathematically like if we take two vectors, we can find another vector with certain properties but why do we use it in physics, if we consider a hypothetical physical quantity like force which is equal to cross product of certain vectors?

For example, the force exerted on a charge in motion in an uniform magnetic field.

Why is it so? Why does that force have to be a cross product of two vectors?

Is it possible to come up with them when what we do is just observe the nature?