It is  well known that   $T\mathbb{S}^{n-1}$ is diffeomorphic to $M= f^{-1}(1)$ where 
$f:\mathbb{C}^n\rightarrow \mathbb{C}$ is $f(z):=\sum_{i=1}^{n} z_{i}^{2}$.
Two questions:

1) Is  $M$  a  symplectic  submanifold of $\mathbb{C}^n\sim \mathbb{R}^{2n}$ (with the standard symplectic structure)?

If  the answer is affirmative, we can consider two  symplectic structures for $T\mathbb{S}^{n-1}$. The first is the original structure of the tangent or  cotangent bundle, the second one is the  pull back  structure of  $M$.

2) Are these structures equivalent?