Density of the linear span of products of harmonic polymomials

Induction on $n$. Base $n=2$ is clear, as you said. Let $\nu$ be a non-trivial finite compactly supported (complex) measure in $\mathbb R^n$ orthogonal to any product $uv$. Then we can smear it a bit and get a non-trivial continuous compactly supported function $f$ orthogonal to each product. Now choose any $n-1$ dimensional space and consider the harmonic functions that do not depend on the orthogonal variable. Applying the induction assumption, we see that the integral of $f$ along every line must be $0$ (otherwise we can project $f$ to get something non-trivial in $\mathbb R^{n-1}$). This is enough to conclude that the Fourier transform of $f$ is identically $0$, so $f$ must be $0$.