Proof without using Green's Identity

Question

I am a TA for an Analysis II (i.e. Multidimentional Mathematical Analysis) class and the professor recently left a question to show $$\int_{\partial \Omega} \frac{\partial \Gamma}{\partial \nu} (x-y)dS_y=1$$ for a bounded domain $\Omega \in \mathbb{R}^n$ and any $x \in \Omega$, where $\Gamma$ is the fundamental solution of Laplace equation and $\nu$ denotes the outer normal of $\partial \Omega$. I am clear that this is trivial using Green's second identity by taking one function as $1$ and the other as $\Gamma$ but the Green's Representation has not been introduced by the professor up to now. So I want a proof without using Green's Theorem. Thank you very much!