Was there an arena in ancient Greece and Rome?

Only in ancient Rome. Never heard of it in ancient Greece.

The ancient Greeks and Romans were completely different, with completely different personalities and temperaments.

The ancient Greeks were literati, and the ancient Romans were soldiers of the rough brothers.

Greeks love philosophy and like to argue with others in the street. Greeks like plays and roaming poems. Their amphitheater is not used to watch animal fighting games, but to watch plays and poems.

Romans don't like these. Relatively speaking, the Romans were very "uneducated".

The characteristics of the Romans are a little close to those of Spartans in many ancient Greek city-States.