COMMUTER SMT UNAUTH TKOF. FLEW OVER ACR-MLG CROSSING RWY DOWN FIELD.

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WE WERE #4 OR 5 FOR TKOF ON RWY 22R BOS. AFTER A SHORT WAIT OF 4 OR 5 MINS; WE WERE TAXIIED INTO POS TO HOLD; AS ANOTHER COMPANY FLT WAS CLRED FOR TKOF. THE F/O WAS FLYING AND I HAD THE RADIOS. WE WENT INTO POS TO HOLD; AND I SET MY HDG BUG FOR 290 DEGS; WHICH WE HAD RECENTLY BEEN RECEIVING. AS WERE GOING INTO POS I HEARD THE TWR CLR ACR Y ACROSS RWY 22R AT THE DEP END. THE NEXT XMISSION FROM THE TWR WAS ACTUALLY FOR COMPANY RSO (ACCORDING TO THE TWR'S TAPES; WHICH I HAVEN'T HEARD) AND WENT SOMETHING LIKE THIS: 'COMPANY RSO TURN RIGHT 250; CONTACT DEP.' WHAT WE HEARD WAS: 'COMPANY GBO TURN RIGHT HDG 250; HERE WE GO;' OR WORDS TO THAT EFFECT. I RESET MY HDG BUG AT THE SAME TIME; SO THIS IS HOW I KNOW I REPLIED. HOWEVER; THAT XMISSION WASN'T ON THE TAPES. IN ANY CASE MADE A NORMAL DEP; AND AS WE WERE PASSING MIDFIELD AT AN ALT OF 200-300'; I NOTICED AN ACR Z MLG XING DOWNFIELD; WHICH I RPTED TO THE TWR. TWR PROMPTLY ADVISED ME THAT I HADN'T BEEN CLRED FOR TKOF; BUT TO CONTACT DEP. I PHONED MY CHIEF PLT AND THE FAA TWR SUPVR UPON ARR IN WATERVILLE; ME; AND WAS INFORMED THAT THE TWR DIDN'T THINK THIS WAS VERY SERIOUS BECAUSE NO REAL HAZARD ACTUALLY OCCURRED; BUT THAT BECAUSE THEY WERE IN THE MIDDLE OF AN ATC EVALUATION; A RPT WOULD BE FILED. I FEEL THAT THIS INCIDENT OCCURRED DUE TO MY INEFFECTIVE SUPERVISION OF THE F/O. ALTHOUGH I AM A NEW ATP; I AM VERY WELL AT HOME IN THE ATC ENVIRONMENT; HAVING BEEN A CTLR FOR ALMOST 11 YRS (MIL AND PVT CONTRACT); AND THE HOLDER OF A COMMERCIAL PLT CERTIFICATE FOR ALMOST 10 YRS. WE WERE RESTED; HAVING RPTED XX15 XX30 THE 2 DAYS PREVIOUS; WITH 36 HRS OFF PRIOR TO THAT. ON THE DAY OF THIS INCIDENT WE RPTED EARLY AND THIS WAS OUR SECOND LEG OF THE MORNING. I FEEL EVERYONE WANTS TO BE A HELPFUL CAPT; AND AS SUCH SOME PEOPLE MAY BE RELUCTANT TO CRITICIZE OR CORRECT SUBORDINATES. THIS MAY HAVE BEEN A FACTOR. I NOW REALIZE THAT NO MATTER WHAT THE SITUATION I; AS THE ATP CAPT; AM IN COMMAND OF THE FLT AND SHALL TAKE WHATEVER STEPS ARE NECESSARY IN THE FUTURE TO ENSURE ITS SAFE CONDUCT (EVEN IF THIS MEANS VERIFYING ATC INSTRUCTIONS WHEN I'M NOT SURE AND LOOKING FOOLISH IF I'M WRONG).