Subject: Why is the cross product defined the way it is (Gyroscope)

Every explanation I have ever seen of why a gyroscope works the way it 
does starts with a definition of the cross product.  Included in this 
definition is the fact that the cross product of 2 vectors, a & b, is 
another vector that is perpendicular to a & b and obeys the right-hand 
rule.  I have never seen a justification for this definition (except 
occassionally in math texts the justification is that it explains a 
gyroscope).  My question is this:  Is there any "real" reason for the 
definition of the cross product or was it defined simply to coincide with 
our experience of nature?  If it is defined so that it agrees with our 
experience, is there any way to see why it was defined this way without 
reference to a gyroscope?

Re: Why is the cross product defined the way it is (Gyroscope)