I am trying to prove the following special case of the hodge decomposition theorem in differential geometry for a n component vector field $V_i$ in $\mathbb{R}^n$.
any vector can be written as the following combination $V_i = −\partial_i \phi + \epsilon^{ii_2 i_3···i_n}\partial_{i_2}F_{i_3i_4···i_n}$
where $F$ is a rank $n-2$ anti-symmetric tensor.
I have read the proof of the Helmholtz theorem, but I don't know how I can generalize it. Do I have to assume some Ansatz? What would that be? It would be great if somebody could show a stepwise method showing tensor calculations, and stating the necessary theorems?