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.