F[x_,y_,z_] := Det[(({{f[x], g[x], h[x]}, {f[y], g[y], h[y]}, {f[z], g[z], h[z]}}))]

F[x,y,z]*F[u,v,z]-F[x,u,z]*F[y,v,z]+F[x,v,z]*F[y,u,z] // Simplify

Sum[
G[x[1],x[2],x[3]]G[x[4],x[5],x[6]] /. Thread[Array[x, 6] -> p],
{p, Permutations[Array[x,6]]}
]

Factor[
Sum[
F[x[1],x[2],x[3]]F[x[4],x[5],x[6]] /. Thread[Array[x, 6] -> p],
{p, Permutations[Array[x,6]]}
]
]

Factor[
Sum[
F[x[1],x[2],x[3]]Signature[p] /. Thread[Array[x, 3] -> p],
{p, Permutations[Array[x,3]]}
]
]

Factor[
Sum[
F[x[1],x[2],x[3]]F[x[4],x[5],x[6]] /. Thread[Array[x, 6] -> p],
{p, Permutations[Array[x,6]]}
]
]

Factor[
Sum[
Times[
F[x[1],x[2],x[3]]F[x[4],x[5],x[6]] /. Thread[Array[x, 6] -> p],
Signature[Take[p,3]]Signature[Take[p,-3]] 
],
{p, Permutations[Array[x,6]]}
]
]