Solving a system of equations involving smooth functions

Question

This is a follow-up question of my previous question. Suppose $h_{i\overline{j}}$, where $1\leq i, j\leq n$, are functions defined on $\mathbb{C}^n$ such that $h_{i\overline{j}}$ are smooth when viewed as functions defined on $\mathbb{R}^{2n}$ and $H=(h_{i\overline{j}})$ is a Hermitian positive definite matrix. I wonder if it is always possible to find smooth functions $p_1,..., p_n$ defined on $\mathbb{R}^{2n}$ such that $$\frac{\partial p_i}{\partial z_j}+\overline{\frac{\partial p_j}{\partial z_i}}=h_{i\overline{j}}\mbox{ and } \frac{\partial p_i}{\partial \overline{z}_j} -\frac{\partial p_j}{\partial \overline{z}_i}=0.$$ In the previous question, I require $p_i$ to be holomorphic, which seems to be too strong.