You can get dimensional hardwood from the big box stores, but it is usually thinner stock and rather expensive. Generally, most hardwoods are sold as rough stock. If you go to a hardwood dealer you will find what you need. It will be sold by the board foot with a thickness in quarters of an inch. For example, 8 quarters will be 2 inches thick. The problem is that you will need the equipment to mill that material into the 2x6 stock that you desire. As a, somewhere in the middle, option, if you have a Woodcraft store near you, they will usually have the wood that you are looking for and can possibly mill it down for you. The downside is that it will be a fair amount more expensive that the typical hardwood dealer.
As kh54s10 said, you may just want to use some regular lumber and then glue a thin red oak facade onto it.