Consider the following weak form of Darboux's theorem:
Let $\omega$ be a symplectic form defined on a neighborhood of $0$ in $\mathbb{R}^{2n}$, meaning that $d \omega =0$ and $\omega$ has no kernel as a skew symmetric form on the tangent space at $0$. Then, passing to a possibly smaller neighborhood $U$ of $0$, there are coordinates $x_1, \ldots, x_n$, $y_1, \ldots, y_n$ on $U$ such that $\omega = dx_1 \wedge dy_1 + dx_2 \wedge dy_2 + \cdots + dx_n \wedge dy_n$ on $U$.
Proof: Make a linear change of coordinates so that $\omega(0)$ is $du_1 \wedge dv_1 + du_2 \wedge dv_2 + \cdots + du_n \wedge dv_n$. Define $\omega_0$ to be the differential form $du_1 \wedge dv_1 + du_2 \wedge dv_2 + \cdots + du_n \wedge dv_n$ and set $\omega_t = t \omega + (1-t) \omega_0$. We will use this family of differential forms.
So all of the $\omega_t$ give the same skew symmetric form at $0$. Also, $d \omega$ and $d \omega_0$ are both $0$. Shrinking our neighborhood of the origin to be contractible, by Poincare's theorem, we have $\omega = d \theta$ and $\omega_0 = d \theta_0$ for some one forms $\theta$ and $\theta_0$.
All of the differential forms $\omega_t$ are nondegenerate at $0$. Shrinking our neighborhood, we can arrange that they are nondegenerate everywhere on $U$. Thus, there is a vector field $X_t$ such that $\omega_t(X_t, Y) = \theta(Y) - \theta_0(Y)$ for any vector $Y$.
Flowing along $X_t$ defines an automorphism of a neighborhood of $0$. One can compute that this flow pulls back $\omega_0$ to $\omega_t$. So the pullbacks of the coordiante functions $u_i$ and $v_i$ are our desired $x_i$ and $y_i$.
