  Antisymmetric Dzyaloshinskii-Moriya (DM) interactions generating from the
spin-orbit coupling induce various fascinating properties, like magnetoelectric
(ME) effect, weak ferromagnetism and non-trivial topological spin textures like
skyrmions, in real materials. Compared to their symmetric isotropic exchange
counterpart, these interactions are generally of a weaker order of strength,
creating modest twisting in the spin structure which results in weak
ferromagntism or weak linear ME effect. Our proposed two-sublattice model, in
contrast, predicts a hitherto unobserved, charge ordered non-collinear
ferrimagnetic behavior with a considerably high magnetization $\textbf{M}$
coexisting with a ferroelectric (FE) order with an electric polarization
$\textbf{P}$ and a strong cross coupling between them which is primarily driven
by the inter-sublattice DM interactions. The key to realize these effects is
the coupling between these microscopic interactions and the FE primary order
parameter. We predict microscopic mechanisms to achieve electric field
$\textbf{E}$ induced spin-reorientation transitions and 180$^{\circ}$ switching
of the direction of $\textbf{M}$. This model was realized in the hexagonal
phase of LuFeO$_3$ doped with electrons. This system shows $P \sim$ 15
$\mu$C/cm$^2$, $M \sim$ 1.3 $\mu_B$/Fe and magnetic transition near room
temperature ($\sim$ 290 K). Our theoretical results are expected to stimulate
further quest for energy-efficient routes to control magnetism for spintronics
applications.
