With $A_X$ the complex of $\mathbb{R}$-differential forms on $X$, the Künneth theorem states that \begin{align*} A_X \otimes A_Y &\to A_{X \times Y}, \\ (\omega,\eta) &\mapsto {\rm pr}_X^\ast \omega \wedge {\rm pr}_Y^\ast \eta \end{align*} gives an isomorphism on cohomology. Is there any explicit formula describing a quasi-inverse of this map? I.e. a map $ A_{X\times Y} \to A_X \otimes A_Y $ inducing an inverse on cohomology groups ?
Thanks