  We give a generalization of the concept of near-symplectic structures to 2n
dimensions. According to our definition, a closed 2-form \omega on a
2n-manifold M is near-symplectic, if it is symplectic outside a submanifold Z
of codimension 3, where \omega^{n-1} vanishes. This extends the concept known
in dimension 4. We depict how this notion relates to near-symplectic
4-manifolds and broken Lefschetz fibrations via some examples. We define a
generalized broken Lefschetz fibration, or BLF, as a singular map with
indefinite folds and Lefschetz-type singularities. We show that given such a
map on a 2n-manifold over a symplectic base of codimension 2, then the total
space carries such a near-symplectic structure, whose singular locus
corresponds precisely to the singularity set of the fibration. A second part
studies the geometry around the codimension--3 singular locus Z. We describe a
splitting property of the normal bundle N_Z that is also present in dimension
four. A tubular neighbourhood for Z is provided, which has as a corollary a
Darboux-type theorem for near-symplectic forms.
