This paper presents an attitude estimation methodology for robotic platforms equipped with a set of rate gyros that can implicitly measure the Earth's rotation rate. As is well known, standard strategies for attitude determination rely on explicit body-fixed measurements of at least two inertial reference vectors. In this work, the Earth's spin is regarded as one of two constant inertial reference vectors. However, because that spin is only implicitly measured, we first implement a Kalman filter to obtain an estimate of it. This filter also yields: unbiased estimates of the second inertial reference vector, which is explicitly measured in the body-fixed frame; and estimates of two biases related to the sensor measurements. The platform's rotation matrix is then computed using a Kalman-based observer evolving on the manifold $SO(3)$ that is driven by the output of the first filter. Realistic simulation results are presented to showcase the performance of the proposed technique in two different scenarios.