F[x_,y_,z_] := Det[(({{x[1], x[2], x[3]}, {y[1], y[2], y[3]}, {z[1], z[2], z[3]}}))];
dot[x_, y_] := x[1]y[1]+x[2]y[2]+x[3]y[3];
Symmetrize[vars_, expr_] := Sum[expr /. Thread[vars -> p], {p, Permutations[vars]}];
AntiSymmetrize[vars_, expr_] := Sum[Signature[p]*expr /. Thread[vars -> p], {p, Permutations[vars]}];

Symmetrize[{a,b,c}, f[a,b,c]]

Factor[Symmetrize[{a,b,c,d,e,f}, F[a,b,c]F[d,e,f]]]

Factor[Symmetrize[{a,b,c}, F[a,b,c]F[a,b,c]]]

Factor[AntiSymmetrize[{a,b,c},dot[a,v1]dot[b,v2]dot[c,v3]]]

Factor[AntiSymmetrize[{a,b,c,d},dot[a,v1]dot[b,v2]dot[c,v3]dot[d,v4]]]

r1=Factor[Symmetrize[{a,b,c,d}, F[a,b,c]dot[a,y1]dot[b,y2]dot[c,y3]]]

r2=Factor[Symmetrize[{a,b,c, d}, F[a,b,c]dot[a,y1]dot[b,y2]dot[c,d]dot[d,y3]]]

Simplify[r2/r1]

Factor[Symmetrize[{a,b,c, d,e,f}, F[a,b,c]dot[a,e]dot[e,y1]dot[b,f]dot[f,y2]dot[c,d]dot[d,y3]]]

res1=Expand[Symmetrize[{a,b,c,d,e,f},
F[a,b,c]F[d,e,f]v1[a]v2[b]v3[c]v4[d]v5[e]v6[f]
]]

res2=Expand[Symmetrize[{a,b,c,d,e,f},
dot[a,d]dot[b,e]dot[c,f]AntiSymmetrize[{d,e,f}, 
v1[a]v2[b]v3[c]v4[d]v5[e]v6[f]
]]]

res1-res2

res3=Expand[Symmetrize[{a,b,c,d,e,f},
F[d,e,f]F[d,e,f]dot[c,f]v1[a]v2[b]v3[c]v4[a]v5[b]v6[c]
]]

First /@ {res1, res2,res3}