I'm interested if there is the explicit forms of basis functions in $L^2(S^n), n\geq 3$.

For $n=1, n=2$ basis functions are well known: $\{e^{ik\phi}\}_{k\in\mathbb{Z}}$, $\{p^{|m|}_n(\cos \gamma) e^{im\phi} | (\gamma,\phi)\in S^2, n \geq 0, m = \overline{-n,n}\}$. Also the following fact is known: \begin{equation} L^2(S^n) = \bigoplus_{k=0}^{\infty} H_k(S^n), \end{equation} where $H_k(S^n)$ - space of harmonic, homogeneous of degree $k$ polynomials on $S^n$. Actually, space $H_k(S^n)$ is obtained just by restricting $H_k(R^{n+1})$ to $S^n$: \begin{equation} \pi : H_k(R^{n+1})\rightarrow H_k(S^n): P(x)\mapsto P\left( \frac{x}{\|x\|} \right), \end{equation} which is also a bijection. As space $H_k(R^{n+1})$ is finite-dimensional, one can choose linearly independent system in $H(R^{n+1})$ and after mapping to sphere linear independence will preserve. So it is left to do the orthogonalization process and that's it for $H_k(S^n)$. This is a straight method to look for the exact basis, however it involves huge calculations.

Maybe someone did it before and there is a reference for the exact basis functions?

p.s. The question is motivated by calculating the integral: \begin{equation} I(\rho,\theta) = \int\limits_{S^n} Y_{k}^{i}(\omega) e^{i\rho (\omega,\theta)} \, d\omega, \end{equation} where $Y_{k}^i$ is a spherical harmonic function from basis in $H_k(S^n)$.

(most of the info I took from the book Stein&Weiss -- Introduction to Fourier Analysis in Euclidean spaces).