Cross product with basis vectors different from i, j, and k A student just asked me about the possibility of a cross product defined in terms of a nonstandard basis, that is, don't use i, j, and k.
Can anyone point me to an article explaining such a question, if it exists?

Yes, there is. See this great discussion and the paper suggested by @Bye_World. Also, if you are just dealing with $\mathbb{R}^n$, then any vector $a,b$ has a representation under a 'non-standard basis', and so you could compute $a,b$ under that non-standard basis and then use those basis vectors instead. 
