Suppose $f\colon \mathbb{R}^m\times\mathbb{R}^m\to\mathbb{R}^n$ is a bilinear transformation. How do I define and construct the matrix representation of $f$ with respect to the canonical bases of the vector spaces involved? How do I represent in matrix form the action of $f$ on the pair $(u,v)\in\mathbb{R}^m\times\mathbb{R}^m$?
This is not a homework. I'm self studying this topic and can't find this construction in the books I have at my disposal.
On
When $f:{\Bbb R^n}\times{\Bbb R^n}\to\Bbb R$ the answer is $$[f]=[f(e_i,e_j)],$$ where the $e_i$ are the canonical base vectors.
Now if $f:{\Bbb R^n}\times{\Bbb R^n}\to\Bbb R^2$ then $f(v,w)=(f_1(v,w),f_2(v,w))$ with both $f_1,f_2$ bilinear so a pair of matrices can be associate as: $$[f_1]=[f(e_i,e_j)]\quad \mbox{and}\quad [f_2]=[f_2(e_i,e_j)]$$ Now you can figure it up what is next.
I would say you can't! You will need three indices such an array!
If $n=1$, you can use Gramian matrices. For other $n$, the only thing I can think of is creating one of those for each of the $n$ components.
You could of course just as well make a "canonical application" $c:\mathbb{R}^{2m} \to\mathbb{R}^m\times\mathbb{R}^m$ and work with $c\cdot f$ isntead but I don't think that's what you are looking for