functional analysis – Under what conditions is the operator $ – sum partial_ {x_i} a_ {ij} partial_ {x_j} $ self-adjoint?

To define:
$$
mathcal {L} = – sum_ {i, j} a_ {ij} partial_ {x_i} partial_ {x_j},
$$

or $ a_ {i, j} = a_ {j, i} $. Yes $ a_ {i, j} = delta_ {i, j} $ the operator is the Laplacian who is known to be self-adjoint in the $ L ^ 2 $ standard.
For other choice of coefficients $ a_ {i, j} $ – is the operator $ mathcal {L} $ self assistant in the $ L ^ 2 $ standard?

If we limit our attention to the $ mathbb {R} ^ $ 3 case, we have this
$$
int_ Omega v mathcal {L} u , d vec {x} = int _ { partial Omega} v (A nabla u) cdot hat {n} dS – int_ Omega ( A nabla u) cdot nabla vd vec {x},
$$

$$
int_ Omega mathcal {L} v , d vec {x} = int _ { partial Omega} u (A nabla v) cdot hat {n} dS – int_ Omega (A nabla v) cdot nabla ud vec {x},
$$

or $ v in C_c ^ { infty} ( Omega) $, $ A = (a_ {i, j}) $. Symmetry of $ A $, the second term in the two equalities is identical, but is there a reason for the first term to be equal in both equalities?