Exponential map for symplectic manifolds

Question

Let $(M, \omega)$ be a symplectic manifold. Darboux's theorem states that, for any point $p \in M$, there exists a neighborhood $U$ of $p$, a neighborhood $V \subseteq \mathbb{R}^{2n}$ of $0$, and a symplectomorphism $(U, \omega) \cong (V, \omega_\mathrm{std})$. I'm curious whether the following stronger statement would be true: does there exist an "exponential map" $$ \exp : T M \to M $$ where for all $p \in M$, the map $$ \exp_p := \exp|_{T_p M} : (T_p M, \omega_p) \to (M, \omega) $$ preserves the symplectic structure in a neighborhood of the origin? (In other words, $\exp_p$ restricts to a Darboux chart for all $p \in M$.)