In his book Mathematical Methods of Classical Mechanics, V.I. Arnold writes
To each vector $\xi$, tangent to a symplectic manifold $(M^{2n},\omega^2)$ at the point $\mathbf{x}$, we associate a 1-form $\omega^1_\xi$ on $TM_\mathbf{x}$ by the formula $$\omega^1_\xi(\boldsymbol{\eta})=\omega^2(\boldsymbol{\eta},\xi)\quad\forall\boldsymbol{\eta}\in TM_\mathbf{x}$$
I see how $\omega^2$ furnishes an isomorphism $\xi\rightarrow \omega^1_\xi$. But then Arnold has the example
In $\mathbb{R}^{2n}=\{(\mathbf{p},\mathbf{q})\}$ we will identify vectors and 1-forms using the Euclidean structure $(\mathbf{x},\mathbf{x})=\mathbf{p}^2+\mathbf{q}^2$. Then the correspondence $\xi\rightarrow\omega^1_\xi$ determines a transformation $\mathbb{R}^{2n}\rightarrow \mathbb{R}^{2n}$.
By "Euclidean structure" I presume he is talking about the Euclidean metric. But I don't see how this isomorphism induces the transformation $\mathbb{R}^{2n}\rightarrow \mathbb{R}^{2n}$ or furthermore how to determine the matrix of this transformation.
And help would be greatly appreciated.