The French are not exactly a skating nation, but they do have some exceptional skating locations. There is the first floor of the Eiffel Tower, for example. It is a bit small, but cozy. Ice-skating at an altitude of 57 meters with a view to die for. Especially when you go at night and the city lights are on, it is not something you can do just anywhere.
If you have a ticket to go up the tower, you will also be allowed to go skating. You are not allowed to bring your own skates. You must rent them on the spot. It draws many tourists, also people who have never done it before. Look for the opening hours on the website.
Another famous, much larger skate rink, is the skating rink in front of the prestigious Hôtel de Ville, the Paris Town Hall.
More information: www.toureiffel.paris