In the book by Dusa McDuff and Salamon, there is the following corollary:
Corollary 2.4 Suppose that $\omega_t$ is a smooth family of nondegenerate skew-symmetric bilinear forms on $\mathbb{R}^{2n}$ depending on a parameter $t$. Then there exists a smooth family of matrices $\Psi_t$ \in $\mathbb{R}^{2n \times 2n}$ such that $\Psi_t^*\omega_t=\omega_0$ for every $t$.
Proof: Theorem $2.3$ and Gram-Schmidt.
Theorem $2.3$ is the existence of a symplectic basis on a symplectic linear space $(V,\omega)$.
However, I don't know how we put together this theorem and Gram-Schmidt in order to furnish corollary 2.4. How is this done?