What is an example of an orientable compact $2n$ dimensional manifold $M$ whose all even dimensional De Rham cohomology groups $H\_{\mathrm{DeR}}^{2i}(M)$ are nonzero, but $M$ does not admit any symplectic structure?


**Added:** As it is indicated in the comments, this post is not a duplicated post.



An orientable compact even dimensional manifolds whose all even cohomologies do not vanish but it does not admit any symplectic structure
For your $n=2k$, $\mathbb{C} P^n\#\mathbb{C} P^n$ does not even admit an almost complex structure, so it cannot be symplectic.


See also:
1) Goertsches-Konstantis' paper *"Almost complex structures on connected sums of complex
projective spaces"* which answers the following MO question  

2) [Does $\Bbb{CP}^{2n} \# \Bbb{CP}^{2n}$ ever support an almost complex structure?](https://mathoverflow.net/questions/216272/does-bbbcp2n-bbbcp2n-ever-support-an-almost-complex-structure)


Including Miller's comment for the remaining (your $n=2k+1$) dimensions: $S^2\times(S^4\times\cdots\times S^4)$


