The tensor ω for the standard flat 4D non degenerate anti-symmetric closed bilinear form is:
0 1 0 0
-1 0 0 0
0 0 0 1
0 0 -1 0
It might seem more natural to say that a form is closed if all boundaryless 2D submanifolds have zero area.
The property “closed” is intended, however, as a local property of the form.
Consider the 4D torus which is the Cartesian product of 4 circles.
It is intended to consider this a symplectic manifold.
Use the antisymmetric form given above in the natural torus coordinates.
Consider the 2D submanifold {(0, 0, x, y) | 0≤x≤π & 0≤y≤π}.
This submanifold has no boundary but its total area is 4π2.
It has no boundary but is not the boundary of any 3D submanifold.