  We consider the operator $L=-{\rm div}(A\nabla)$, where the $n\times n$
matrix $A$ is real-valued, elliptic, with the symmetric part of $A$ in
$L^\infty(\mathbb{R}^n)$, and the anti-symmetric part of $A$ only belongs to
the space $BMO(\mathbb{R}^n)$, $n\ge2$. We prove the Gaussian estimates for the
kernel of $e^{-tL}$, as well as that of $\partial_t^le^{-tL}$, for any
$l\in\mathbb{N}$. We show that the square root of $L$ satisfies the $L^p$
estimates $\left\Vert{L^{1/2}f}\right\Vert_{L^p}\lesssim\left\Vert{\nabla
f}\right\Vert_{L^p}$ for $1<p<\infty$, and $\left\Vert{\nabla
f}\right\Vert_{L^p}\lesssim\left\Vert{L^{1/2}f}\right\Vert_{L^p}$ for
$1<p<2+\epsilon$ for some $\epsilon>0$ depending on the ellipticity constant
and the BMO semi-norm of the coefficients. Finally, we prove the $L^p$
estimates for square functions associated to $e^{-tL}$. In another article of
the authors, these results are used to establish the solvability of the
Dirichlet problem for elliptic equation ${\rm div}(A(x)\nabla u)=0$ in the
upper half-space $(x,t)\in\mathbb{R}_+^{n+1}$ with the boundary data in
$L^p(\mathbb{R}^n,dx)$ for some $p\in (1,\infty)$.
