Just look at n-sphere. A lecture note from Stony Brook is also available.
You can find it in Fock's paper (Fock, V. (1935).)or in some recent papers- like (Howard, S. "Fundamental Solution of Laplaces Equation in Hyperspherical Geometry.") or (Jing-Jing, F., Ling, H., & Shi-Jie, Y. (2011). Solutions of laplace equation in n-dimensional spaces. Communications in Theoretical Physics, 56(4), 623.) for advanced studies.