Let $K$ denote the Kelvin transform, and let $|\alpha|:=\sum_{j=1}^n\alpha_j$ denote the weight of the multi-index $\alpha\in\mathbb{N}^n$. Then, an explicit base for the space of homogeneous harmonic polynomials in $n$ variables and degreee $m$, $\mathcal H^m:=\mathcal H^m[x_1,\dots ,x_n]$, is
$$\big\{ K\big(\partial^\alpha\|x\|^{2-n}\big) : |\alpha|=m, \alpha_n\leq 1 \big\}.$$
Indeed, an easy combinatorial computation shows it has the right cardinality $$\operatorname{card}\{\alpha\in\mathbb{N}^n : |\alpha|=m, \alpha_n\leq 1 \}=
{n+m-1\choose n-1} - {n+m-3\choose n-1} = \operatorname{dim}\mathcal H^m,$$
(the latter dimension being already known from linear algebra considerations on the operator $\Delta$). On the other hand, one can verify that it spans the same linear space as the analogous set without the constraint on $\alpha_n$:
$$\{ K\big(\partial^\alpha\|x\|^{2-n}\big): |\alpha|=m\},$$
which is the whole space $\mathcal H^m$. So it is a base.

A nice (and free) reference for these classical facts is *Harmonic Function Theory*, by S.Axler, P.Bourdon, and W.Ramey; see thm 5.25.