You can either claim the compensation yourself from the airline or ask someone to assert it, such as a lawyer or an online provider focused on passenger rights such as "flightright" or "faiplane".
It can be advantageous if you first ask the airline for payment. If they do not respond or refuse payment even though you are entitled to a compensation payment, the airline is in default and you can then hire a lawyer to enforce your claim. The lawyer has to pay the airline in this case!
If you yourself want to ask the airline to pay, no special form is required. Many airlines already offer ready-made online forms. You can use these, but they do not have to. You can also ask the airline for example by fax, email or letter for payment.
Tip: You should set the airline a deadline for the payment of, for example, 3 weeks in order to put the airline in default. You should also make sure that you have proof that the airline has received your request for payment, eg insertion letter or fax delivery report.