When I saw "n" in the title I assumed they wanted something that worked reasonably for large n. Rejection is no good because of the notorious "curse of dimensionality". So my idea was to choose a suitable distribution on the radius, then draw from it and choose the angles at random (not sure what those angles are called). You might have to delete the point at the center for that to work.
The article explicitly discusses both. Versions for low dimensions that have favorable properties, and those for high dimensions.
The method you propose is not used in high dimensions in practice as it would involve evaluating order n^2 trigonometric functions and is also far harder to implement than the methods discussed in the article.