Fourier transform in $\mathbb{R}^n$ of $e^{-\|x\|}$

Question

Given $n\in\mathbb{N}$, I'm stuck in the calculation of the Fourier transform in $\mathbb{R}^n$ of $$(x_1,\dots,x_n)\mapsto\exp(-\sqrt{x_1^2+\dots+x_n^2}).$$ I know how to compute the result for each $n$ odd (see e.g. [1]: Fourier transform of $e^{-|x|}$), but also in this case I don't know a closed formula and, for $n$ even, I don't know how to procede. Can anyone give a closed formula in $n\in\mathbb{N}$ and show how to calculate it? Thanks in advance.