Let $V$ be the vector space of $\mathbb{C}^\infty$ functions from $\mathbb{R}^n$ to $\mathbb{R}^n$ and consider some linear map $L:V\rightarrow V$ such that $L$ can be expressed strictly in terms of first order derivatives. Also, $L$ should be invariant under isometries of $\mathbb{R}^n$, that is, if $\forall x\in\mathbb{R}^n, f_2(x)=f_1(Ax+b)$ and $g_2(x)=g_1(Ax+B)$ for some vector $b$ and matrix $A$ such that $A^TA=I$, then $L(f_1)=g_1\iff L(f_2)=g_2$. For example, the curl map $f\rightarrow \nabla\times f$ is such a map. Notice these maps are closed under linear combination and thus form a vector space that I will call $W_n$.
I would like a basis for $W_n$ for each $n$. For example, for $n=1$, the derivative operator can be taken to be the single element of such a basis. Further more, I suspect $W_2$ is $0$ dimensional (please let me know if I am wrong here). For $3$ dimensions, I only know curl to have these properties and so I suspect $dim(W_3)=1$. For $W_n$ with $n≥4$, I have no idea.
Finally, I suspect this question can be better phrased (especially the isometry invariance condition) so if you know how to rephrase it better and would like to, go ahead. Also the title could probably be rephrased.