homework and exercises - Finding a 'vector potential' such that $mathbf E = nablatimes mathbf C$ for a point charge

Supposedly, "Any divergence-free vector field can be expressed as the curl of some other divergence-free vector field" over a simply-connected domain.

So, what is one such vector potential which works for half of the Coulomb field?

To be clear, I want a vector potential whose curl equals the vector field $\mathbf R/|\mathbf R|^3$ for $z>0$ (for any $x$ and any $y$). $\mathbf R$ is the position vector $(x,y,z)$.

I know the scalar potential method is usually used instead of this, but am curious about how ugly a vector potential would look. If this gets answered, it should then be easy to answer this.