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$\omega_2(SO(n))=?$, that is, What is the second stiefel whitney class of SO(n)?

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What vector bundle did you have in mind? –  Grumpy Parsnip Apr 21 '11 at 16:52
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up vote 6 down vote accepted

$SO(n)$ is a Lie group and so its tangent bundle is trivial. Thus $w_2(TSO(n)) = 0$.

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Thank you for your help. Furthermore, Is the Frame bundle of a spin manifold as a manifold spin? –  Boh Liu Apr 22 '11 at 4:57
    
@Boh Liu : You should ask this as a separate question, as it has little to do with the original one. –  Raeder Apr 22 '11 at 18:04
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