  We define the Conway skein module C(M) of ordered based links in a 3-manifold
M. This module gives rise to C(M)-valued invariants of usual links in M. We
determine a basis of the Z[z]-module C(F x [0,1])/Tor(C(F x [0,1])) where F is
the real projective plane or a surface with boundary. For cylinders over the
Moebius strip or the projective plane we derive special properties of the
Conway skein module, including a refinement of a theorem of Kawauchi and
Hartley about the Conway polynomial of strongly positive amphicheiral knots in
S^3. We also determine the Homfly and Kauffman skein modules of F x [0,1] where
F is an oriented surface with boundary.
