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
.