Given an integral $$I=\int\limits_{\mathbb{R}^n} \cdot \; dx,$$ we can introduce polar coordinates, such that $$I=\int\limits_{\Bbb S^{n-1}} \cdot \; d\theta.$$ Another way to express the latter one is $$\int\limits_{\|x\|=1} \cdot\; dx.$$
Is there an adequate notation, to express the integral in polar coordinates, $$\int\limits_{\Bbb S^{n-1}} \cdot\; d\theta,$$ such that the integral's variable refers to $x$ and the integral domain makes clear we are integrating on $\mathbb{R}^n$?
I would suggest using some notation like
$$\int_{\mathbb{R}^n} f(x) dx = \int_0^\infty \int_{\mathbb{S}^{n-1}} r^{n-1} f(rx) dx dr$$
or if you want to keep the $f(x)$ even
$$\int_{\mathbb{R}^n} f(x) dx = \int_0^\infty \int_{r\mathbb{S}^{n-1}} f(x) dx dr $$