While reading the book Introduction to the $h$-Principle by Y. Eliashberg and N. Mishachev, I noticed that the authors state, at the end of section 9.1.A, that the space of all symplectic structures on Euclidean space $\mathbb R^{2n}$, which they identify as $\text{GL}(2n,\mathbb R)/\text{Sp}(2n)$, is not contractible, whereas the space of all Euclidean structures of the aforementioned space, which can be identified with $\text{GL}(2n,\mathbb R)/\text{O}(2n)$, is contractible.
How can one prove such assertions? I am not familiar with homotopy theory at all, so I am totally clueless on how to proceed.
Thanks in advance for your answers.
Edit: as the comments suggest, it is easy to prove that the space of symplectic structures is not contractible. However, in the book, it is asserted that this space has exactly two connected components, and that both are not contractible. So, how can we prove these statements?