symplectic sructure on a ball

Question

We know by Darboux theorem that any symplectic form on a manifold $W^{2n}$ is locally symplectomorophic to the standard symplectic form $dx\wedge dy$ on $R^{2n}$. Is it true that any symplectic form on a ball $B^{2n}$ of arbitrary radius is symplectomorphic to the standard one? or -I guess equivalently- can we extend this local symplectomorphsim as long as we stay on a ball (say handles of different indices in a Morse decomposition of the manifold) in the manifold $W^{2n}$?