Endorsing a skill without being prompted in LinkedIn:
Visit the profile page of the person whom you want to endorse.
Scroll the screen down and find the Skills & Expertise section. Here you will find the identified skills of that person.
If you find an individual skill that you want to endorse, then place your mouse over the + sign found next to the skill you want to endorse and click Endorse.
If you have endorsed any skills already then they would appear with a blue filled in + sign near the skill name. And you will know that you have endorsed that as your profile picture would be underlined to the right side of the skill name.
In case you want to remove a particular endorsement, then use your mouse and hover over the blue + sign > then click Remove Endorsement link.
This way you will be able to add more skill endorsements any time you want.