Modulo redefinition of the phases of operators, projective representations are in correspondence with central extensions (as both are built out of nontrivial algebraic 2-cocycles). For n>2, Spin(n) is a universal cover, so the phase of any of its projective representations is a coboundary, which is to say that operators can be redefined to make the representation linear.
tl;dr: There are no nontrivial projective representations of Spin.
What happens in n=1,2? I haven't thought about it.
By the analysis of Wigner and especially Bargmann, Spin(3) of QM is isomorphic to SU(2) which is known to be semi-simple, hence its Lie algebra has no non-trivial central extensions. This implies that the projective representations of SU(2) are trivially related to its linear representations.
It is true that in semi-simple lie algebra one can always remove the central charges by a redefinition of generators, but sometimes it is possible to remove the central charges even if the algebra is not semi-simple as is the case with Poincare group by using some special argument.
For n>2, we have our spin groups as double cover of the SO(n) and they are simply connected, hence they have no nontrivial central extension. However this is not the case for n≤2, like for n=2 the group is U(1) and it is infinitely connected.