Suppose $F_{i_1\cdots i_n}(x)$ is a totally symmetric rank $n$ tensor field in 3 dimensions ($i=1,2,3$). Is it possible to decompose $F_{i_1\cdots i_n}(x)$ in a way which generalizes the Helmholtz decomposition of a smooth, rapidly decaying vector field $F_{i}(x)$?
$$F_i(x) = \phi_i(x) + \partial_i\phi(x)$$
where $\phi_i(x)$ is transvere: $\partial^i\phi_i(x)=0$. My guess for such a generalization would be
$$F_{i_1\cdots i_n}(x) = \phi_{i_1\cdots i_n}(x) + \partial_{(i_1}\phi_{i_2\cdots i_n)}(x) + \partial_{(i_1}\partial_{i_2}\phi_{i_3\cdots i_n)}(x) + \cdots + \partial_{i_1}\cdots\partial_{i_n}\phi(x) $$
where all $\phi_{i_1\cdots i_k}(x)$ for $k=1,\dots, n$ are totally symmetric for rank 2 and higher and transverse: $\partial^{i_1}\phi_{i_1\cdots i_k}(x)=0$, and the parentheses surrounding indices denotes total symmetrization.
I am at a loss for how to try to prove this. It is not clear to me that the standard proof of Helmholtz decomposition can be adapted for higher rank tensors. Any hints on how to approach this?