closed differential 2-forms

Question

Let $\psi$ be a fixed differential 1-form on $\mathbb{R}^n$. I'm looking for an emplicit differential 1-form on $\mathbb{R}^n$, $\phi$, such that $\psi\wedge \phi$ is closed. If $\psi=\sum^n_{i=1}A_i\,dx^i$ and $\phi=\sum^n_{i=1}B_i\,dx^i$, then $$\psi\wedge \phi=\sum^n_{i,j=1}A_iB_j\,dx^i\wedge dx^j$$ and the condition $d(\psi\wedge \phi)=0$ implies $${\partial A_i \over \partial x^k}B_j+A_i{\partial B_j \over \partial x^k}+{\partial A_k \over \partial x^j}B_i+A_k{\partial B_i \over \partial x^j}+{\partial A_j \over \partial x^i}B_k+A_j{\partial B_k \over \partial x^i}=0. $$ My question would be solved if I can solve this differential equation for $B_1,...,B_n$ ($A_1,...,A_n$ are given) but I can't. Does someone know the solution to this differential equation? or a way to solve it? or another way to answer to my question?