You can take omega-3 and vitamin E together, and they are often sold combined for a sensible reason. The one situation where the pairing deserves a second thought is if you also take a blood thinner.
Why they are paired
The fats in fish oil are delicate and prone to going rancid, which is what causes the stale, fishy aftertaste of an old bottle. Vitamin E is an antioxidant, and a small amount of it protects those fragile fats from oxidising. That is why a lot of fish oil products already include some vitamin E as a stabiliser, and why the two are a natural pairing rather than a clash. For most people, taking them together is fine and there is no competition between them.
The one caveat: blood thinners
Both omega-3 at higher doses and vitamin E at higher doses have a mild blood-thinning effect of their own. On their own this is minor. Taken together, and especially on top of a prescription anticoagulant or antiplatelet drug, the effect can add up in a way worth flagging to your clinician, particularly before any surgery or dental work. If you are not on a blood thinner, this is not something to worry about at normal supplement doses.
How to take them
Take both with a meal that contains some fat, since both are fat-soluble and absorb better that way. There is no need to separate them; the whole point of the pairing is that they go together. As with all fish oil, the form on the label affects how much you actually absorb.
To check omega-3 or vitamin E against any medication you take, and especially against a blood thinner, the free checker will show you the result and the reasoning. The wider question of which supplements clash is covered in what not to take together.