[susan53]

Hi Beatrix,
Two things to consider here :
a) this is a defining relative clause ( it specifies which dress I've bought)
b) the relative pronoun serves as object in the relative clause (it replaces "the dress" in the proposition "she is wearing the dress")

So :

a) As it's a defining clause, in British English either that or which could be used. American friends tell me that using which is frowned upon in Am.Eng and that even in Brit.Eng the Guardian style guide recommends the use of that. Personally, I prefer "which" but it's just a matter of stylistic preference.
In some Brit.Eng varieties of the language what could also be used, but that's non-standard, so I would only teach it receptively if students were going to be in an area where it might be used. It wouldn't be accepted in a test.

b) As the relative pronoun is the object, it could (as you say) also be omitted. This would be usual in spoken English, but it would be more likely to be included in more formal written text.

In addition...

c) "as" sounds a bit odd here - though in spoken English when there's little time for planning might well occur as a sort of blend of two grammatical ideas. However, it would normally be used when the verb in the second clause was the same as that in the first - eg I'm wearing the same dress as she is (wearing). / I've bought the same dress as she has (bought).