We consider divergence form elliptic operators L = − d i v A ( x ) ∇ L= {-}\mathrm {div}\, A(x) \nabla , defined in the half space R + n + 1 \mathbb {R}^{n+1}_+ , n ≥ 2 n\geq 2 , where the coefficient matrix A ( x ) A(x) is bounded, measurable, uniformly elliptic, t t -independent, and not necessarily symmetric. We establish square function/non-tangential maximal function estimates for solutions of the homogeneous equation L u = 0 Lu=0 , and we then combine these estimates with the method of “ ϵ \epsilon -approximability” to show that L L -harmonic measure is absolutely continuous with respect to surface measure (i.e., n-dimensional Lebesgue measure) on the boundary, in a scale-invariant sense: more precisely, that it belongs to the class A ∞ A_\infty with respect to surface measure (equivalently, that the Dirichlet problem is solvable with data in L p L^p , for some p > ∞ p>\infty ). Previously, these results had been known only in the case n = 1 n=1 .