  We develop the Seiberg-Witten map using the gauge-covariant star product with
the noncommutativity tensor $\theta^{\mu\nu}(x)$. The latter guarantees the
Lorentz invariance of the theory. The usual form of this map and its other
recent generalizations do not consider such a covariant star product. We
construct the Seiberg-Witten map for the gauge parameter, the gauge field and
the strength tensor to the first order in the noncommutativity parameter
$\theta^{\mu\nu}(x)$. Prescription for the generalization of the map to higher
orders is also given. Interestingly, the associativity of the covariant star
product both in the first and second orders requires the same constraints,
namely, on the $\theta^{\mu\nu}(x)$ and on the space-time connection. This fact
suggests that the same constraints could be enough to ensure the associativity
in all orders. The resulting Seiberg-Witten map applies both to the internal
and space-time gauge theories. Comparisons with the Seiberg-Witten map based on
other (non-covariant) star products are given and some characteristic
properties are also presented. As an application, we consider the $GL(2, C)$
noncommutative gauge theory of gravitation, in which it is shown that the
connection determines a space-time with symplectic structure (as proposed by
Zumino et al [AIP Conf. Proc. 1200 (2010), 204, arXiv:0910.0459]). This example
shows that the constraints required for the associativity of the
gauge-covariant star product can be satisfied. The presented $GL(2, C)$
noncommutative gauge theory of gravitation is also compared to the one (given
by Chamseddine [Phys. Rev. D 69 (2004), 024015, hep-th/0309166]) with
non-covariant star product.
