Complement of a point in $ S^n $ is homeomorphic to $ \mathbb{R}^n $

Question

Define $ S^n $ to be all the $ (n + 1) $-tuples in $ \mathbb{R}^{n + 1} $ whose Euclidean distance to the origin equal $ 1 $. I read in here that the complement of a point in $ S^2 $ is homeomorphic to $ \mathbb{R}^2 $ via the stereographic projection. Does this hold for every $ n \geq 2 $, i.e. the complement of a point, say $ (0, 0, \dots, 0, 1) $, is homeomorphic to $ \mathbb{R}^n $? I am having trouble finding an explicit homeomorphism here. Also if $ U $ and $ V $ are complement of a point such that $ S^n \subseteq U \cup V $, then is $ U \cap V $ homeomorphic to $ S^n $?