Does the set of harmonic functions $u\in C^2(\Omega)$ on $\Omega$ form a proper cone?

Question

Let $\Omega$ be a domain in $\mathbb{R}^n$.

Subharmonic functions have the following property of a cone: if $u$ and $v$ are subharmonic in $\Omega$ and $a, b\in\mathbb{R}_+$ then $au+bv$ is also subharmonic in $\Omega$.

I was reading about proper cones one time and I've started wondering, does any $f\in C^2(\Omega)$ have a representation as a difference $f = u-v$ with $u, v\in C^2(\Omega)$ subharmonic in $\Omega$?