Symplectic structures on the tangent bundle of a smooth manifold

We give a method to lift \((2,0)\)-tensors fields on a manifold \(M\) to build symplectic forms on \(TM\). Conversely, we show that any symplectic form \(\Om\) on \(TM\) is symplectomorphic, in a neighborhood of the zero section, to a symplectic form built naturally from three \((2,0)\)-tensor fields associated to \(\Om\).