We know in dimension $3$, \begin{align} \partial_{i}= \frac{x_i}{r} \partial_{r} - \varepsilon_{ijk} \frac{x^j}{r} \frac{R^k}{r} , \end{align} where $\varepsilon_{ijk}$ are Levi-Civita symbols and \begin{align} x_i:=\delta_{ij}x^j , \quad R^k:= \varepsilon^{klq} x_l\partial_{q} . \end{align}
I wonder if there is an analog in higher dimension $n$ (for $n \geq 4$) that can also be written in a simple form like this one?
The formula you gave in the beginning is a special case of the vector triple product formula in $\mathbb{R}^3$ $$ a\times (b\times c) = (a\cdot c) b - (a\cdot b) c $$ Let $v$ be an arbitrary vector in $\mathbb{R}^3$, and let $x$ be the vector field $(x_1, x_2, x_3)$, then you have $$ x\times (x\times v) = (x\cdot v) x - \|x\|^2 v $$ or $$ v = \frac{1}{\|x\|^2} \left( (x\cdot v) x - x\times (x\times v) \right) $$ You can check that your object $R$ is just the vector field $x\times v$.
The same identity holds (almost) in arbitrary dimensions.
In $\mathbb{R}^n$ with $n > 2$, let $$ R^{a_1a_2\ldots a_{n-2}} = \varepsilon^{a_1a_2\ldots a_{n-2} b c} x_b \partial_c $$ Then using the product formulae for Levi-Civita symbols, $$ \varepsilon_{mna_1 a_2\ldots a_{n-2}}x_nR^{a_1a_2\ldots a_{n-2}} = (n-2)! \delta_{mn}^{bc} x^n x_b \partial_c = (n-2)! (x_m x^c \partial_c - |x|^2 \partial_m) $$
And so you have, finally, $$ \partial_m = \frac{x_m}{r} \partial_r - \frac{1}{(n-2)! r^2} \varepsilon_{mna_1 a_2\ldots a_{n-2}}x_nR^{a_1a_2\ldots a_{n-2}} $$
Personally, though, I don't like the formula using the Levi-Civita symbol too much. It is much easier to write $$ \Omega_{ij} := x_i \partial_j - x_j \partial_i $$ from which you get $$ x^i \Omega_{ij} = r^2 \partial_j - x_j r \partial_r $$ or $$ \partial_j = \frac{x_j}{r} \partial_r + \frac{x^i}{r^2} \Omega_{ij} $$ which works in any dimension. (This is also the form you will find it most frequently in the literature on wave equations.)
