Can I use a cheeseboard as a cutting board?
You certainly can use a "cheese board" or other single-slab piece of suitable wood as a chopping board. 

What you might want to consider is that there is a reason why even expensive, good-quality chopping boards are made from multiple pieces instead of one big slab:
Wood can warp and / or develop cracks if exposed to changes in humidity or when drying out. Using multiple pieces glued together in alternating directions counters that effect to some degree. This does not mean that your one-piece idea won't work, but that you should take good care of the board and avoid long exposure to water (don't soak it) or dry heat. Keep it well-oiled, clean and towel dry it immediately after use and leave it in a well-ventilated area between uses.