Matrix Representation of the Tensor Product of Linear Maps
$f$ has matrix $A$ with respect to $\{e_1, e_2\}$. Hence $f(e_1) = a^i_1e_i$, analogously $g(e_1) = b^j_1e_j$. By bilinearity of $\otimes$, therefore \[ f(e_1) \otimes g(e_1) = a^i_1b^j_1 (e_i \otimes e_j) \] So if say, you decide that $(e_1 \otimes e_1, e_1 \otimes e_2, e_2 \otimes e_1, e_2 \otimes e_2)$ is your ordered basis of $\mathbb R^2 \otimes \mathbb R^2$, then the first column of your matrix is $(a_1^1b_1^1, a_1^1b_1^2, a_1^2b_1^1, a_1^2b_1^2)^t$.
Check out http://en.wikipedia.org/wiki/Kronecker_product