Putnam 1986 A5

Suppose $f_1(x), f_2(x), \dots, f_n(x)$ are functions of $n$ real variables $x = (x_1, \dots, x_n)$ with continuous second-order partial derivatives everywhere on $\mathbb{R}^n$. Suppose further that there are constants $c_{ij}$ such that $$\frac{\partial f_i}{\partial x_j} - \frac{\partial f_j}{\partial x_i} = c_{ij}$$ for all $i$ and $j$, $1\leq i \leq n$, $1 \leq j \leq n$. Prove that there is a function $g(x)$ on $\mathbb{R}^n$ such that $f_i + \partial g/\partial x_i$ is linear for all $i$, $1 \leq i \leq n$. (A linear function is one of the form $$a_0 + a_1 x_1 + a_2 x_2 + \cdots + a_n x_n.)$$