Now the decision: To skin ot not to skin. Most people will skin their fillets, because flatfish are typically cooked in either refined ways where skin would be a hindrance, or fried, where it would curl up the fillet and cause it to cook poorly.
The exceptions are really large flatfish such as big turbot and halibut, or, if you are ever so lucky, an Atlantic fluke larger than 12 pounds. These fillets are actually better cooked by steaking them out into fillet steaks, with a strip of skin left on.
Alternatively, teeny flatfish such as sand dabs are also wonderful fried whole (but gutted) with the skin on. But then you would not be filleting them, would you?
The way to skin a fillet is to anchor it firmly at the tail end with one hand, then slipping the knife along the skin. This is where a true fillet knife earns its keep: It is far more flexible than most other blades, so you can actually maintain pressure on the knife as you cut off the skin -- the knife will bend. It can take some effort, so keep a firm hold on that tail end!