A metric naturally arise from the Euclidean symplectic structure?

For $$n>1$$ let $$\omega=\sum_{i=1}^n dx_i\wedge dy_i$$ be the standard symplectic structure on $$\mathbb{R}^{2n}=\mathbb{R}^n \times \mathbb{R}^n$$. We define the following distribution $$D$$ on $$\mathbb{R}^{2n}\setminus\{0\}$$:

For $$Z\in \mathbb{R}^{2n}\setminus\{0\}$$ we define $$D_Z=\{V\in \mathbb{R}^{2n}\mid \omega(V,Z)=0\}$$

This is a nonintegrable distribution of codimension $$1$$. We define a meteic on $$\mathbb{R}^{2n}\setminus\{0\}$$ as follows: The distance $$d(x,y)$$ is the infimum of the Euclidean length of all $$D$$- horizontal curves joining(connecting) $$x$$ to $$y$$.

Is this metric well defined(i.e. is this distribution totally non integrable)?Does this metric arise from a Riemannian metric on $$\mathbb{R}^{2n}\setminus\{0\} \}$$?