I know that the Brew Potion feat can be used to make these items, but can I also use skills?
The Paizo site states that potions can be crafted with the Brew Potion feat, Spellcraft skill, and Craft (alchemy) skill.
Does this mean I can just have Craft (alchemy) and not take the feat Brew Potion and still be able to make potions and oils? Or do I still need the feat?
http://paizo.com/prd/magicItems/magicItemCreation.html
create magic items, spellcasters use special feats which allow them to invest time and money in an item's creation. At the end of this process, the spellcaster must make a single skill check (usually Spellcraft, but sometimes another skill) to finish the item. If an item type has multiple possible skills, you choose which skill to make the check with. The DC to create a magic item is 5 + the caster level for the item. Failing this check means that the item does not function and the materials and time are wasted. Failing this check by 5 or more results in a cursed item (see Cursed Items for more information).
Note that all items have prerequisites in their descriptions. These prerequisites must be met for the item to be created. Most of the time, they take the form of spells that must be known by the item's creator (although access through another magic item or spellcaster is allowed). The DC to create a magic item increases by +5 for each prerequisite the caster does not meet. The only exception to this is the requisite item creation feat, which is mandatory. In addition, you cannot create potions, spell-trigger, or spell-completion magic items without meeting their spell prerequisites.
You can use spellcraft or craft(alchemy) for the final check to create the potion/oil, but the brew potion feat is still required.
The link you cite doesn't quite say that. What the Creating Potions section says is that Brew Potion is required, plus either Spellcraft or Craft (alchemy). So yes, you still need the feat.
Like others said, you definitely need Brew Potion feat to create magical potions. However, I allowed non-magical alchemicals (including oil) to be created through craft(alchemy) skill in my game. You can see some examples of non-magical creations here, together with the Craft DC.
There is also a calculator that you might want to take a look at. It helps players and GMs keep track of crafting checks and you can see what you need to craft an alchemical item.
