Quotient space of harmonic functions on punctured plane

Question

Let $a_1,\ldots,a_n$ be $n$ distinct points in $\mathbb{C}$ and let $\Omega := \mathbb{C} \setminus \{a_1,\ldots,a_n\}$. Define $H(\Omega)$ to be the space of harmonic functions on $\Omega$ and $R(\Omega)$ the subspace of $H(\Omega)$ consisting of real parts of holomorphic functions on $\Omega$. I want to show that the equivalence classes of $$\log\vert z - a_1 \vert,\ldots, \log \vert z - a_n \vert$$ form a basis for the quotient space $H(\Omega)/R(\Omega)$. I know how to show linear independence. I have seen rather lengthy arguments that the above functions span the space, but I am wondering if there is a way to show that the dimension of $H(\Omega)/R(\Omega)$ must be $n$ without actually computing a basis. That way I can conclude the above functions form a basis based on their linear independence.