By Darboux lemma we may assume that the symplectic form has the form
   
      ω = dx^1 ⋀ dx^3 + dx^2 ⋀ dx^4
      
      in local coordinates (x^1, x^2, x^3, x^4). Then
      
      ω = 1/2 OmegaForm(i,j) dx^i ⋀ dx^j.

OmegaForm = {
{0,0,1,0},
{0,0,0,1},
{-1,0,0,0},
{0,-1,0,0}
};

OmInv=Inverse[OmegaForm];
OmInv//MatrixForm

(* Xi is an arbitrary 2-form. *)
XiNorm[a_,b_]:= Signature[{a,b}] ξ[Min[{a,b}],Max[{a,b}]]

(* evaluate
    dx^i1 ⋀ ... ⋀ dx^i4( d/dx^k1 ⋀ ... ⋀ d/dx^k4)
*)
Perm[i1_,i2_,i3_,i4_,k1_,k2_,k3_,k4_]:= Module[
{Ivec,Kvec,a,b},
Ivec={i1,i2,i3,i4};
Kvec={k1,k2,k3,k4};
Det[Table[KroneckerDelta[Ivec[[a]],Kvec[[b]]],{a,1,4},{b,1,4}]]
]

Evaulate traditional definition in coordinates for a 2-form on a 4-manifold. 
See Libermann and Marle: Symplectic geometry and analytic mechanics, p. 43

Evaluate 1/2 ω vedge ω( ξ^sharp wedge d/dx^k3 ⋀ d/dx^k4)

defTrad=Table[
Sum[1/16 OmegaForm[[i1]][[i2]] OmegaForm[[i3]][[i4]] XiNorm[l,m] OmInv[[l]][[k1]]OmInv[[m]][[k2]] 
Perm[i1,i2,i3,i4,k1,k2,k3,k4],
{i1,1,4},{i2,1,4},{i3,1,4},{i4,1,4},
{l,1,4},{m,1,4},{k1,1,4},{k2,1,4}
],
{k3,1,4},{k4,1,4}
];
defTrad//MatrixForm

Definition in paper:

   ast(xi)_ij = -1/2 ω^ab xi_ab ω_ij   - xi_ij

defPaper =Table[
-1/2Sum[OmInv[[a]][[b]] XiNorm[a,b],{a,1,4},{b,1,4}] OmegaForm[[i]][[j]]-XiNorm[i,j],
{i,1,4},{j,1,4}
];
defPaper = Simplify[defPaper];
defPaper //MatrixForm

Check that the two definitions coincide

Simplify[defPaper==defTrad]

Note: The above computation also gives an explicit expression for the star operator in Darboux coordinates.