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Also, if i'm not mistaken, steenrod gives a more direct argument in topology of fibre bundles, but he might be using the long exact sequence of a fibration (which you mentioned). I've found lots of different proofs that so (n) is path connected, but i'm trying to understand one i found on stillwell's book naive lie theory. It's fairly informal and talks about paths in a very

Six year from now, the old man's age will be only three times that his first born son. Find age of each. The question really is that simple: Prove that the manifold $so (n) \subset gl (n, \mathbb {r})$ is connected. It is very easy to see that the elements of $so (n