u = {u1, u2, u3}; v = {v1, v2, v3}; w = {w1, w2, w3};

u\[Cross]v

u\[Cross]v == -v\[Cross]u

(u . (v\[Cross]w)        // Expand   )==Det[(({{u1, u2, u3}, {v1, v2, v3}, {w1, w2, w3}}))]

u\[Cross](v\[Cross]
            w) == ((u . w) v - (u . v) w   // 
          Expand)

Clear[u, v, w, z]

z = u\[Cross]v

zLength = (Sqrt[z[[1]]^2 + z[[2]]^2 + z[[3]]^2])

uLength = (Sqrt[u[[1]]^2 + u[[2]]^2 + u[[3]]^2])

vLength = (Sqrt[v[[1]]^2 + v[[2]]^2 + v[[3]]^2])

Expand[(zLength^2)]

Expand[(uLength^2) (vLength^2) - Dot[u, v]^2]

Expand[(zLength^2)] == 
      Expand[(uLength^2) (vLength^2) - (((u . v))^2)]