Let $X$ and $Y$ be complex manifolds with local coordinates $(x_1, ..., x_n)$ and $(y_1, ..., y_m)$, respectively. The exterior algebra $\Lambda^2(X \times Y)$ decomposes as $$\Lambda^2(X \times Y) = \Lambda^2(X) \oplus (\Lambda^1X \otimes \Lambda^1Y) \oplus \Lambda^2(Y).$$ Therfore, a smooth $2$-form on $X \times Y$ may written as \begin{eqnarray*} \omega &=& \sum_{i,j=1}^n \varphi_{ij} dx_i \wedge dx_j + \eta + \sum_{k,\ell =1}^m \psi_{k\ell} dy_k \wedge dy_{\ell}, \end{eqnarray*}
where $\eta \in \Omega^1(X) \otimes \Omega^1(Y)$. I'm not sure how to write $\eta$ in local coordinates. Any help is appreciated.