"I am too excited to sleep!"
Translation:Ik ben te opgewonden om te slapen!
It can be omitted with certainly auxiliary verbs. Usually the construction also contains "te". See here for more examples: http://www.dutchgrammar.com/en/?n=Verbs.Au11
In this sentence, the construction is "too X to do Y". You can never leave out the Dutch "om" that case. The same is true when it means "in order to".
- Ik lees om slimmer te worden - I read (in order) to become smarter.