  Building on recent progress in constructing derivations on Fourier algebras,
we provide the first examples of locally compact groups whose Fourier algebras
support non-zero, alternating 2-cocycles; this is the first step in a larger
project. Although such 2-cocycles can never be completely bounded, the operator
space structure on the Fourier algebra plays a crucial role in our
construction, as does the opposite operator space structure.
  Our construction has two main technical ingredients: we observe that certain
estimates from [H. H. Lee, J. Ludwig, E. Samei, N. Spronk, Weak amenability of
Fourier algebras and local synthesis of the anti-diagonal, Adv. Math., 292
(2016); arXiv 1502.05214] yield derivations that are "co-completely bounded" as
maps from various Fourier algebras to their duals; and we establish a twisted
inclusion result for certain operator space tensor products, which may be of
independent interest.
