v[x_, y_, z_, t_] := {{0, a x + b, c x + d}, {g z^2 + h z + m, d z^2 + e z + f, 0}, {f[t], g[t], h[t]}, {a x z, a y z, -a z^2}, {-Sin[Pi y] Exp[-Pi^2 t/R], Sin[Pi x] Exp[-Pi^2 t/R], 0}, {(a/k) x, - (a/k) y + n k D[y z, z], -n k D[y z, y]}, {-((2 x)/(c^2 Re)) - (d y)/(x^2 + y^2), -((2 y)/(c^2 Re)) + (d x)/(x^2 + y^2), (4 z )/(c^2 Re)}}
v[x, y, z, t] // MatrixForm

VG = Table[Simplify[Grad[v[x, y, z, t][[i]], {x, y, z}]], {i, 7}];
S = Table[Simplify[1/2 (VG[[i]] + Transpose[VG[[i]]])], {i, 7}];
W = Table[Simplify[1/2 (VG[[i]] - Transpose[VG[[i]]])], {i, 7}];
Do[Print[VG[[i]] // MatrixForm, S[[i]] // MatrixForm, W[[i]] // MatrixForm], {i, 7}]