ACR ADVTECH LGT ON FRT OP MAKES INCORRECT TAXI AT OMA; SUFFERS TAXIWAY EXCURSION TRYING TO CORRECT.

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ON TAXI OUT OF THE CARGO AREA AT OMA; I INADVERTENTLY TAXIED ONTO A TAXIWAY THAT WAS CLOSED. I MADE A R TURN ONTO 'G' INSTEAD OF A L TURN. I IMMEDIATELY STOPPED THE ACFT; THEN TURNED BACK INTO THE CARGO RAMP AT THE NEXT TAXIWAY. UPON ENTERING THE CARGO RAMP THE L MAIN GEAR OF THE LGT CUT ACROSS A SMALL AREA OF GRASS DURING THE TURN TO MY L. BECAUSE OF THE PROX TO SEVERAL SMALL ACFT; I THOUGHT IT WOULD BE SAFEST TO SHUT THE ACFT DOWN AND GET A TUG TO PUSH THE ACFT BACK TO THE MAIN CARGO AREA. OUR MECH INFORMED ME WE HAD RUN OVER AN AREA OF GRASS. WE MADE AN ENTRY INTO THE ACFT LOGBOOK. THE ACFT WAS INSPECTED AND THERE WAS NO DEBRIS OR DAMAGE. THERE WAS NO DAMAGE TO ARPT PROPERTY. AFTER OUR MECH SIGNED THE LOGBOOK THE ACFT WAS CAREFULLY PUSHED BACK AND WE DEPARTED. I OFFER TWO SUGGESTIONS TO PREVENT THIS IN THE FUTURE. 1) A FEEL MY OWN COCKPIT RESOURCE MGMNT WAS INADEQUATE AND THERE WAS NOT ENOUGH CREW COM PRIOR TO TAXI-OUT. A CRM PROGRAM IS BEING IMPLEMENTED AND I AM CERTAIN I CAN IMPROVE MY SKILLS. 2) ATC GND CTL DID NOT GIVE SPECIFIC INSTRUCTIONS TO TAXI TO RWY 35. WE WERE JUST CLRED TO TAXI TO RWY 35. WITH CONSTRUCTION AROUND THE AREA AT R; IT WOULD HAVE BEEN HELPFUL TO BE INFORMED TO TAXI TO RWY 35 VIA A L TURN ON 'G'; R TURN ON 'I' ETC. THIS WOULD HAVE 'KEYED' ME TO THE SITUATIONAL AWARENESS I WAS NEEDING.