  Let $B$ be an arrangement of linear complex hyperplanes in $C^d$. Then a
classical result by Orlik \& Solomon asserts that the cohomology algebra of the
complement can be constructed from the combinatorial data that are given by the
intersection lattice.
  If $B'$ is, more generally, a $2$-arrangement in $R^{2d}$ (an arrangement of
real subspaces of codimension $2$ with even-dimensional intersections), then
the intersection lattice still determines the cohomology {\it groups} of the
complement, as was shown by Goresky \& MacPherson. We prove, however, that for
$2$-arrangements the cohomology {\it algebra} is not determined by the
intersection lattice. It encodes extra information on sign patterns, which can
be computed from determinants of linear relations or, equivalently, from
linking coefficients in the sense of knot theory. This also allows us (in the
case $d=2$) to identify arrangements with the same lattice but different
fundamental groups.
