In the case of $\mathbb{R}^{2n}$ the exponential map just becomes $$ exp_{p}(v) = p + v$$ which results in $$ Dexp_{p}(w)_{v} = w $$ and proves that this map is a symplectomorphism.
I am wondering if this is also the case in a general symplectic manifold and in the likely event that it isn't true what possible properties are necessary to prove it is a symplectomorphism.