what are you using to test? test client also affects results.
verified it is using N-speeds?
quick and dirty test, using wired clients, not wireless clients:
connect computer on one end
connect known gigabit capable device on the other end. can be gigabit switch, router etc.
ensure the patch cables are cat5e or better
connect both devices. ensure computer can see the other device
assuming windows, in adapter settings, force it to 1Gbps speed.
if connection maintains, highly likely it is cat5e or better (cat 5 *may* be able to maintain 1Gbps with short cable length)
if using a switch, if the indicator shows 1Gbps link speed, most likely it is cat5e or better
or, remove the jack face plate. there is usually some slack. if you are lucky, that cable length will have have the markings to show the cable rating
edit: RT-N56U has gigabit ports. the 300Mbps is the wireless speed rating:
https://www.asus.com/us/Networking/RTN56U/specifications/
