No because "et dyr" would have to mean "an expensive". Here dyr is a noun, expensive is an adjective. It is det.
From what I've been learning you can't. It's like the german articles "der" "die" das"... you have to memorize them for every noun as you learn it. I've learned here that 75% is "en"... so in doubt I'd try "en". However there seem to be general patterns that an user kindly put here and can help picking the right article:
en, den and denne are for the common* gender.
et, det and dette are for the neuter gender.
de and disse are for the plural, which doesnt have gender
*masculine and feminine genders merged into the common in a lot of germanic languages, including dutch (partially), danish, english (along with the neuter), swedish and norwegian (mostly)