I was trying to show $\mathbb{C}\otimes_\mathbb{C}\mathbb{C}\cong\mathbb{R}^2$ as $\mathbb{R}$-modules. At one point I would like to prove the existence of an $\mathbb{R}$-module homomorphism from $\mathbb{C}\otimes_\mathbb{C}\mathbb{C}\to\mathbb{R}^2$ by showing there exists a $\mathbb{C}$-balanced map $\mathbb{C}\times\mathbb{C}\to\mathbb{R}^2$, and using the universal property.
One such map is $\psi(a+bi,c+di)=(ac-bd,ad+bc)$. My question is, how would one come up with this map without just trial and error? The term $ac-bd$ looks suspiciously like a determinant, so there must be some reason that this would be a go-to map.