Actually, I would be inclined to look at it the other way- with the cross product of two vectors [b]defined[\b] by and the direction given by the "right hand rule", prove that , but, of course, you can go either way.

In order to show your way, define a "new" product, say *, by , and the "right hand rule". We can easily get the rules , , and as well as , and the general .

Now, for and , multiply "term by term" and use the basic products above to show that this "new" product is precisely the cross product.