It would be the flax seed (linseed) oil you would give to your dog. It has many benefits, but I would suggest mixing it with cottage cheese or yogurt to make the oil water soluble and immediately available. Flax seed and its oil are rich in omega 3 and balanced (for humans, at least) in 3, 6, and 9. It is very good against arthritis, as an anti-inflammatory, and for energy among other things. The flax seed (linseed) itself would need to be freshly ground up, as it goes stale within 15 mins. or so, to get its benefits as otherwise it likely will pass through undigested, but can be added freshly ground to the cottage cheese or yogurt mixture. Be sure to refrigerate the oil, if used as it has a short shelf unrefrigerated.
Coconut oil has many benefits too, and definitely for skin, among many other things because it is rich in lauric acid...a is human mothers' milk...think 'baby skin' softness. You can feed it to your dog, or rub a tiny bit on his skin or both. You will not need much oil to rub on him and most likely he will love the taste. I give coconut oil to my two and, since I use it as a 'lotion' for my skin, they love to lick my hands after I rub coconut oil on either of us.
You could also use emu oil. There are several references to coconut oil on YT...I am not certain which threads at the moment. Keep coconut out of sunlight, but it can be kept at room temp in a dark place or light tight container and is good for frying. It has had many bad and untrue things said about it, unfortunately. My kids love organic eggs cooked in organic virgin coconut oil with a smidgeon of natural ground (pink) salt.
There is much information on these oils available. I do not know about vitamin C oil for humans or dogs. It is my understanding that dogs and cats bodies can mfg vitamin C and should not be given it, whereas humans bodies cannot and must get theirs in their diet, so, if that is true, I do not know what the benefit of vitamin C oil would be for a canine.
Don't know if this helps you or not.
Good luck!