I have been contemplating extending the definition of cross product for matrices, and I wonder if this has been done before.
Basically my definition is, given two 3x3 matrices: $A=(a_{ij})_{i,j=1} ^ 3$ and $B=(b_{ij})$ then $A\times B=(A_i \times B_j)_{i,j=1}^3$ where $A_i=(a_{i1},a_{i2},a_{i3})$ the same with B (just replace A with B a with b and i with j).
Now I checked that it's linear with regard to simple matrix addition $$A\times (B+C)= A\times B + A\times C$$
And obviously it's antisymmetric, I am not sure if there's a nice connection with matrix multiplcation, I mean $A\times (BC)=?$ not sure if there's a simple identity here.
Has this been done already, where may I read on this?
Thanks.