The answer to the question of whether you cook salmon skin up or down is almost always to start skin side down.
The reason for this is that the salmon skin protects the more tender flesh of the salmon during cooking. Once the salmon has started to firm up, you can use a flexible fish turner to flip it over and cook the other side. The secret is to use enough oil in your pan, and to let the fish cook long enough on the skin side for it to hold together when you try and turn it.
The biggest mistake people make is trying to turn their salmon fillets over too soon, resulting in the skin sticking to the pan and tearing. You will actually cook your salmon most of the way through on the skin side before turning it over
Exceptions to the rule
Obvious exceptions to this rule are of course fillets that have had the skin removed, and when you are poaching salmon. If you are poaching it makes no difference which side you start on.