I want to make sure that my proof is correct and see other nice ways how we can show it.
Let $V$ be a symplectic vector space of dimension $2n$ and $W$ is a subspace of $V$ of dimension $2n-1$. We want to show that $W$ is a coisotropic subspace of $V$ i.e. $W^{\omega}\subset W$ where $W^{\omega}=\{v\in V|\omega(v,w)=0\text{ for all }w\in W\}$ with a given symplectic form $\omega$.
Let's take some non-zero vector $v\in V$, and consider a new subspace $W^{\prime}=W\oplus span\{v\}\subset V$. If $W^{\prime}$ is a proper subset of $V$, then we are done. So, it's enough to show that the case $v\notin W$ will gives us a contradiction. But, we will obtain a contradiction by showing that our symplectic form is degenerate since we found a non-zero vector $v\in V$ s.t. $\omega(v,w)=0$ for all $w\in V$.
Does it make sense?