The World Cup quarter-finalists were among the pre-tournament favourites for the continental showpiece but crashed out at the semi-final stage on Wednesday after a 1-0 defeat at the hands of Zambia.
Ghana now must lift themselves for an unwanted third-place play-off against Mali on Saturday, but Dnipro defender Inkoom has promised they will fight for victory in Malabo.
"We need to go all out to get something," he told ghanafa.org.
"It is unfortunate that we could not take the trophy, that was out target but I believe we have to do something for the nation.
"It might be third place but it's another game and important to win for our country.
"We will pick up ourselves and go into the game. We have psyched ourselves up and will do our best."
Ghana beat Mali 2-0 in the group stages but Inkoom does not believe that will have any bearing on Saturday's match.
"It won't be easy," he said.
"They will come for revenge but we will do our maximum best to beat them again."
Ghana will be without four players on Saturday, with captain John Mensah (thigh) and Emmanuel Agyemang-Badu (knee) injured and Derek Boateng and John Boye both suspended.
Mali were forced to make do with a place in the play-off game after they suffered a narrow 1-0 defeat to Ivory Coast in their last-four encounter.
The result continued Mali's horrendous sequence in the African Nations Cup semi-finals that now stands at four successive defeats, and they remain without an appearance in the final since their tournament debut in 1972.