As a node of a wireless sensor network (WSN) with an embedded omnidirectional antenna receives signals emitted from a directional antenna, the received signal strength indication (RSSI) varies with the angle of arrival (AoA) of the received signal. In this paper, we fit RSSI values into a parabola function of AoA between 0° and 90° by quadratic regression analysis. We also set up two directional antennas with perpendicular orientations at the same position and fit the difference of the signal RSSI values of the two antennas into a linear function of AoA between 0° and 90° by linear regression analysis. Based on the RSSI-fitting functions, we propose a novel localization scheme, called ALRD, for a sensor node to fast (within 0.1s) estimate its location with the help of two beacon nodes, each of which consists of two perpendicular-orientationed directional antennas. The fitting functions can easily be stored in a WSN node having limited storage space and their inverse functions can be used to speed up the localization process. We have implemented ALRD and applied it to a WSN in a 10m x 10m indoor square area with two beacon nodes being installed at two ends of an area edge. Our experiments demonstrate that the average localization error is 124 centimeters. We further propose two methods, namely maximum-point-minimum-diameter (MPMD) and maximum-point-minimum- rectangle (MPMR), to reduce localization errors by gathering more beacon signals within 1s for finding the set of estimated locations of maximum density. Such estimated locations are averaged to obtain the final location estimation. The experiment results demonstrate the two methods can reduce the average localization error by a factor about 29% to be 89 centimeters. ALRD allows a sensor node to fast localize itself with small errors and is thus suitable for mobile sensing and actuating applications.