newtonian mechanics - Why is normal force perpendicular?

I realize "normal" is just another word for perpendicular, but semantics aside, what causes normal force to be always perpendicular to the plane?

For example, on an inclined plane, why is it that the plane pushes the box outwards rather than directly against gravity? It seems to me that in order for there to be normal force, there must be a force trying to push the box directly into the plane, but gravity is going directly downwards, not into the plane.