If $Id$ is the $n$ by $n$ identity matrix and $J$ is the $2n$ by $2n$ matrix with $Id$ in the upper right corner and $-Id$ in the lower left corner, then $Sp_{2n} = \{G\in Gl_{2n} : G^{tr}JG = J\}$ is the symplectic group.
The unitary group is the set of matrices $G$ that satisfy that $G^*G = GG^* = Id$. Is there a way to describe the unitary group like how the symplectic is described?
Yes, if you like it: $U(n)=\{G\in Gl_n\colon G^*\, Id\, G=Id\}$.