Let's create some example data:
d = {d1, d2, d3};
A = {{B1, B2, B3}};
d = d - PseudoInverse[A].A.d // ComplexExpand // Simplify;
B = {{0, B3, -B2}, {-B3, 0, B1}, {B2, -B1, 0}};
A.d // Simplify
(* {0} *)
Now, we augment the system matrix B and the right hand side d:
BB = ArrayFlatten[{{B, - Transpose[A]}, {A, 0}}];
dd = Join[d, {0}];
x = LinearSolve[BB, dd][[1 ;; 3]]
(* {(-B3 d2 + B2 d3)/(B1^2 + B2^2 + B3^2), (B3 d1 - B1 d3)/(B1^2 + B2^2 + B3^2), (-B2 d1 + B1 d2)/(B1^2 + B2^2 + B3^2)} *)
Let's test the result:
B.x - d // Simplify
A.x // Simplify
(* {0,0,0} *)
(* {0} *)
With the augmentation, we enforced that $B_1 \, \omega_1 + B_2 \, \omega_2 + B_3 \, \omega_3 = 0$ and we added the orthogonal complement of the image of B to the image of BB.