Places seethe with excitement for this race (12)
I believe the answer is:
'this race' is the definition.
(steeplechase is a kind of race)
'places seethe with excitement' is the wordplay.
'with excitement' indicates an anagram (I've seen this in other clues).
'placesseethe' anagrammed gives 'STEEPLECHASE'.
'for' acts as a link.
(Other definitions for steeplechase that I've seen before include "Race over fences" , "Running event over hurdles and water jumps" , "run?" , "Horse race with ditches and hedges" , "Horse race with jumps and ditches" .)