It is certainly much more uniquely defined, since the direction of the result is determined. But yes, the cross product is defined relative to an orientation an

It is certainly much more uniquely defined, since the direction of the result is determined. But yes, the cross product is defined relative to an orientation and changes coordinates accordingly. That's one of many reasons why the wedge product is more natural. If I could have my way there would not be a single cross product in the entirety of physics.