Will $\vec{f}(x,y,z)=\nabla \times \vec{g}(x,y,z)$ here?

Question

Suppose we have a vector field $\vec{f}(x,y,z)$ defined everywhere in space except a certain region say $(y=3)$. We also know divergence of $\vec{f}(x,y,z)$ is zero at all points where $\vec{f}$ is defined. From this information, is it valid to say that: $$\vec{f}(x,y,z)=\nabla \times \vec{g}(x,y,z)$$ at all points wherever $\vec{f}(x,y,z)$ is defined? Why? Why not?