The camera's sensor is made up of sensels each sensing one of red, green or blue. Placed in a grid of 4 sensels (2-green, 1-red, 1-blue). Conceptually the value of the pixel is the value for each cluster of 4 pixels but the clusters are not unique, the pixel to the right is sharing half of it's sensels with the one to the left - and half with the one above, and half with the one below and half with the one to it's left.
Peek at the picture at: http://www.photozone.de/sensor-types
So you are getting a full resolution picture where the number of pixels in the picture equals the number of sensels on the sensor (less 1 column and 1 row of sensels to account for the edge condition.)
So the short answer is no - given a Bayer sensor, you can not get better resolution than you already are.
Taken to an extreme. If you took a picture of a perfect red wall with your 5Diii - only 5.6 Million red sensels recorded into the raw file would have a value while the remaining 16.7 blue and green sensels would had no value. It's the Bayer algorythm that lets that combination be interpreted as a consistent 23 Megapixel image of a red wall.