CLRNC TKOF NOT ISSUED WITH CREW OF G-IV ADVISED AFTER TKOF MADE AT BAF; MA.

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WE WERE ON AN IFR FLT PLAN FROM BAF TO SAV. I WAS THE CAPT ON THE ACFT WITH THE FO FLYING. THE FO HANDLED THE RADIOS PRIOR TO TKOF AND I HANDLED THE RADIOS AFTER TKOF. AFTER TKOF ON RWY 20 AT BAF; TWR CALLED AND SAID; 'YOU WERE NOT CLRED FOR TKOF.' WE WERE APPROX 500-1000 FT IN THE AIR AT THAT TIME AND I RESPONDED; 'I THOUGHT WE WERE.' TWR'S RESPONSE WAS 'CONTACT DEP CTL ON FREQ.' WE CONTACTED DEP AND THE DEP; CLBOUT; AND FLT PROCEEDED NORMALLY. THERE WAS NEVER ANY TFC CONFLICT. IN FACT; WE NEVER HEARD ANY OTHER TFC ON THE TWR FREQ. THERE WERE NO TCASII CONFLICTS. THE FO AND I REVIEWED THE SIT ON CLBOUT AND WE BOTH THOUGHT THAT WE WERE 'CLRED FOR TKOF.' THE LESSON TO BE LEARNED IS THAT WE MUST BE EXTRA VIGILANT AT ALL TIMES. WE CAN NEVER KNOW WHEN A BRIEF LAPSE MAY PLACE US OR OTHERS IN DANGER. DID WE HAVE A LAPSE AND ONLY THINK WE WERE CLRED FOR TKOF? DID TWR HAVE A LAPSE IN NOT CALLING TO OUR ATTN UNTIL WE WERE WELL INTO THE AIR? WAS THERE IN FACT; 'NO PROB' UNTIL TWR HANDED US OFF TO DEP WITHOUT PRIOR COORD? AT ANY RATE; HUMAN ERRORS WERE MADE AND THERE COULD HAVE BEEN DIRE CONSEQUENCES. THE ONLY CASUALTY IS WOUNDED PRIDE ON MY PART; THE FO'S PART; AND MAYBE THE TWR OPERATOR'S PART. I AM VERY THANKFUL AND FEEL VERY FORTUNATE THAT THERE WAS NOT A CONFLICT; THAT NO ONE WAS HURT; AND THAT NOTHING WAS BENT.