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I have some question on De Rham cohomology: the first one is general. If we calculate De Rham cohomology of a manifold with Mayer-Vietoris sequences we discover that the cohomology is the difect of exactness of the sequence. So can I look at De Rham cohomology as the difficult to extend a local exact form defined on a open set to all manifold?

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Absolutely you can! –  Dylan Wilson Feb 2 '13 at 19:09
    
Do you have a method to calaculate the De Rham cohomology of finite Grassmannians using spectral sequences? –  ArthurStuart Feb 2 '13 at 20:01

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