Needs["Pfaffian`"]
(*4j-3: 2j-1,up*)
(*4j-2: 2j,up*)
(*4j-1: 2j-1,down*)
(*4j: 2j,down*)
L=12(*even*);
A=Table[0,{i,4*L},{j,4*L}];
θ[j_]:=2*Pi*j/L;
Do[
A[[4*j-3,4*j-2]]=Cos[θ[j]];
A[[4*j-1,4*j]]=Cos[θ[j]];
,{j,1,L}];
Do[
A[[4*j-3,4*j-1]]=Sin[θ[j]];
A[[4*j,4*j-2]]=Sin[θ[j]];
,{j,1,L/2}];
Do[
A[[4*j-3,4*j+3]]=Sin[θ[j]];
A[[4*j+4,4*j-2]]=Sin[θ[j]];
,{j,L/2+1,L-1}];
A[[4*L-3,3]]=Sin[θ[L]];
A[[4,4*L-2]]=Sin[θ[L]];
A=N[A-A\[Transpose]];
Pf[A]//N