  We extend the formalisms developed in Gair et al. and Cornish and van
Haasteren to create maps of gravitational-wave backgrounds using a network of
ground-based laser interferometers. We show that in contrast to pulsar timing
arrays, which are insensitive to half of the gravitational-wave sky (the curl
modes), a network of ground-based interferometers is sensitive to both the
gradient and curl components of the background. The spatial separation of a
network of interferometers, or of a single interferometer at different times
during its rotational and orbital motion around the Sun, allows for recovery of
both components. We derive expressions for the response functions of a laser
interferometer in the small-antenna limit, and use these expressions to
calculate the overlap reduction function for a pair of interferometers. We also
construct maximum-likelihood estimates of the + and x-polarization modes of the
gravitational-wave sky in terms of the response matrix for a network of
ground-based interferometers, evaluated at discrete times during Earth's
rotational and orbital motion around the Sun. We demonstrate the feasibility of
this approach for some simple simulated backgrounds (a single point source and
spatially-extended distributions having only grad or curl components),
calculating maximum-likelihood sky maps and uncertainty maps based on the
(pseudo)inverse of the response matrix. The distinction between this approach
and standard methods for mapping gravitational-wave power is also discussed.
