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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). There's a bit of a subtlety here that i'm curious about.can the group of deck transformations be realized as a subgroup of the covering space? The generators of $so(n)$ are pure imaginary antisymmetric $n \\times n$ matrices

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Welcome to the language barrier between physicists and mathematicians I'm particularly interested in the case when $n=2m$ is even, and i'm really only. Physicists prefer to use hermitian operators, while mathematicians are not biased towards hermitian operators

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 I'm not aware of another natural geometric object. It sure would be an interesting question in this framework, although a question of a vastly different spirit. I am really sorry if this answer sounds too harsh, but math.se is not the correct place to ask this kind of questions which amounts to «please explain the represnetation theory of so (n) to me» and to which not even a whole seminar would provide a complete answer

I have known the data of $\\pi_m(so(n))$ from this table I'm looking for a reference/proof where i can understand the irreps of $so(n)$