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.
This is easy enough: $\eta = \sum\limits_{i,k} f_{ik}(x,y) dx_i\wedge dy_k$. (Be careful, by the way. What you're writing is valid for smooth, real manifolds. If you're doing complex manifolds and $x_i$ and $y_k$ are local holomorphic coordinates, then you need not only $dx_i$ but also $d\bar x_i$ and a general $2$-form on a complex manifold is a linear combination of forms of type $(2,0)$, $(1,1)$, and $(0,2)$.)