In the space ${\Lambda }^p$ of polynomial p-forms in ℝⁿ we introduce some special inner product. Let $H^p$ be the space of polynomial p-forms which are both closed and co-closed. We prove in a purely algebraic way that ${\Lambda }^p$ splits as the direct sum $d*\left({\Lambda }^{p+1}\right)\oplus \delta *\left({\Lambda }^{p-1}\right)\oplus H^p$, where d* (resp. δ*) denotes the adjoint operator to d (resp. δ) with respect to that inner product.