How can I get crispy skin when pan frying fish?

You need a hot pan. Use either clarified butter or an oil that can withstand high temperatures without burning. The oil should be quite hot but not at smoking point otherwise the fish closest to the skin will be overcooked and the skin will be charred as well as crispy.

The pan shouldn't be quite as hot when it comes time to cook the fish skin-up.

A good tip is not to move the fish around too much in the pan. Just leave it in place until it should be about ready to turn (You can gently lift a corner of the fish to check but just don't shake it around in the pan very much).

Do not crowd too many pieces of fish if you are frying little pieces. The pan and oil have to be VERY hot. Do not shake the pan.

The thing that gives you a crispy skin is lightly dredging the fillets in seasoned flour. Shake off any excess. I use half regular butter and half olive oil, and add the fillets when the water in the butter has boiled off but before the butter has browned much.

Flip the fillets only once.

As mentioned, don't crowd the pan, as you need any water to escape so that the fish grills, rather than steams.

When done, put the fillets on a plate, then deglaze the pan with butter and lemon juice to make a nice sauce for the fish. You can also add slivered almonds and/or capers.

But the flour is what takes care of the browning/crispiness and it works every time.

The secret I learnt somewhere and can attest from experience to work, is to dry out the skin. Dry the skin with a paper towel. Sprinkle a decent amount of salt on the skin and let it sit for a little while. Dry off the water that was drawn out and wipe off the excess salt. Now proceed to place the fish skin down in preheated oil and let it cook until it's done on that side and ready to be flipped. It should come out very easily, and the skin will still be attached to the flesh.

Start with the fish being nearer to room temperature, rather than straight out of the fridge.
Then get the skin as dry as possible, any water makes crisping hard. You can lightly press with a paper towel and/or a light dust with flour (cornflour, plain flour, potato starch, whatever you like). Next, to a hot pan add some cooking oil that has a high smoking temperature (e.g. rapeseed oil). Lay the fillet/s skin side down away from you to prevent splashbacks. The skin on the fillets tend to want to curl up, so you can lightly press down the flesh so the skin is in contact with the pan. It should flatten and you can leave them to fry skinside down for the majority of the cooking time. The temperature and time will vary depending on the fish and the thickness of the fillet. Once the skin is crispy enough and the heat has travelled to atleast halfway up the flesh of the fish you can turn the fish flesh side down. You can add butter if you like and depending on the heat of the pan and type of fish, (eg sea bass) you could remove from the heat and the remaining heat of the pan can finish cooking the fish. Also you can add lemon juice or any fresh herbs if desired.

TL:DR

Dry Skin, hot pan, fry skin side down for atleast 2/3 of the cooking time, press the skin onto the pan at the start.

How I would do it:

1. Fillet Fish
2. Salt to Taste (Be liberal)
3. Coat with Cornmeal
4. Heat oiled pan, on a low to medium heat
5. Add fish skin side down, turn once skin looks crisp then allow to cook through.