On relation between Solenoidal and irrotational vector fields

Question

Can a solenoidal vector field in $R^3$ be always written as cross product of 2 irrotational vector fields? If this is the case, then given a solenoidal vector field in $R^3$, what is the most efficient way of finding the scalar potentials associated with the underlying 2 irrotational vector fields? To be more precise, given $\vec{v}(x,y,z)$ satisfying $\vec{\nabla}.\vec{v}=0$, what will be the expressions of the scalar functions $H_1(x,y,z)$ and $H_2(x,y,z)$ in terms of the component functions of $\vec{v}(x,y,z)$ so that one has $\vec{v}(x,y,z)=\vec{\nabla}H_{1}\times\vec{\nabla}H_{2}$.