We claim that <w,z>=0 whenever w in W and z in Z.

WAA [a_,b_,i_,j_]:= Signature[{a,b}] Signature[{i,j}]w[Min[{a,b}],Max[{a,b}],Min[{i,j}],Max[{i,j}]]

ZAA [a_,b_,i_,j_]:= Signature[{a,b}] Signature[{i,j}]z[Min[{a,b}],Max[{a,b}],Min[{i,j}],Max[{i,j}]]

Compute <Z,W>

InnProd=Sum[ 
 WAA[p,q,l,m] ZAA[l,m,p,q],
{p,1,4},{q,1,4},{l,1,4},{m,1,4}
]

Obtain constraints for z in Z

Constraints={};

For[i=1, i≤4,i++,
For[j=1, j≤4,j++,
Constraints = Append[Constraints, 
0== 
Sum[ ZAA[i,P,j,P] ,{P,1,4}]
]
]
]

Add constraints for w in W

For[i=1, i≤4,i++,
For[j=i+1, j≤4,j++,
For[a=1, a≤4,a++,
For[b=a+1, b≤4,b++,
Constraints = Append[Constraints, 
Sum[Signature[{i,j,p,q}]WAA[p,q,a,b],{p,1,4},{q,1,4}]
==  
-Sum[ Signature[{a,b,p,q}] WAA[p,q,i,j] ,{p,1,4},{q,1,4}]
]
]
]
]
]

Constraints //MatrixForm

Apply constraints to inner product <Z,W>

Simplify[InnProd, Constraints]