First Isomorphism Theorem for Bilinear Maps

Question

Let $U,V,W$ be three vector spaces over the same field. Let $$\phi:U\times V\to W$$ be a bilinear map. That is, for every $u\in U$ the map $$\phi_u : V\to W$$ $$v \mapsto \phi(u,v)$$ is linear. Likewise for every $v$ in $V$, the map $$\phi_v : U\to W$$ $$u \mapsto \phi(u,v)$$ is linear. Then is there a first isomorphism theorem known is this situation. The obvious guess would be that $$Im(\phi) \cong U\times V / \ker\phi.$$ Although I am having trouble proving it (and am not completely convinced it is true). Is there such a theorem known? Something that relates the image of $\phi$ with its kernel? Any reference or solution would be greatly appreciated.