Putnam 2008 A1

Let $f: \mathbb{R}^2 \to \mathbb{R}$ be a function such that $f(x,y) + f(y,z) + f(z,x) = 0$ for all real numbers $x$, $y$, and $z$. Prove that there exists a function $g: \mathbb{R} \to \mathbb{R}$ such that $f(x,y) = g(x) - g(y)$ for all real numbers $x$ and $y$.