= Solution
The <Symplectic Darboux theorem> says that every point of a $2n$-dimensional <symplectic manifold> has local coordinates $(q^1,\ldots,q^n,p_1,\ldots,p_n)$ in which
$$
\boxed{\omega=\sum_{i=1}^n dq^i\wedge dp_i.}
$$
First choose linear coordinates at the point so the form there is standard. The required <symplectic basis> can be constructed inductively: choose $e,f$ with $\omega(e,f)=1$, split off their span, and repeat on its nondegenerate <symplectic orthogonal complement>. Extend these coordinates to a chart centered at zero.
Let $\omega_0$ be the constant standard form in this chart and $\eta=\omega-\omega_0$. The closed form $\eta$ vanishes at zero. On a small star-shaped ball the radial <Poincare lemma> supplies a primitive
$$
\alpha_x(v)=\int_0^1 t\,\eta_{tx}(x,v)\,dt,\qquad d\alpha=\eta.
$$
Since $\eta_0=0$, this primitive is $O(|x|^2)$. The interpolating forms $\omega_t=\omega_0+t\eta$ are nondegenerate on a common smaller ball, because they all agree with $\omega_0$ at zero and $t$ ranges over a compact interval.
Apply the local version of <Moser's trick>: solve $\iota_{X_t}\omega_t=-\alpha$. The <vector fields> are $O(|x|^2)$ and fix zero. On a sufficiently small ball their flows exist for $0\leq t\leq1$ and remain inside the coordinate chart; the quadratic bound makes their displacement smaller than the available margin. The same pullback calculation gives $f_1^*\omega=\omega_0$. Thus $f_1$ is a local <symplectomorphism> from the standard ball into $M$, and its inverse supplies the desired <Darboux chart>. \b[There are no local symplectic invariants beyond dimension.]
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