It is possible to get a uni-polar voltage from a Wheatstone bridge. If the measurement side (variable side fixed resistor) is placed at one end of the measurement range and you use an isolated supply for bridge excitation, you can ground one side of the bridge output and thus have a 0 to whatever output voltage, center the fixed resistor and have a bi-polar output. If the output voltage caused by the excitation is too high at the extreme resistance difference lower the drive.
The really interesting is to use an AC voltage, two matched resistors and a capacitor for the "reference' component and measure reactance by balancing the bridge.
This is a simplified description. There are better ones here for both Wheatstone and Hay bridges and the math to go with it.
Bob