No, it's not.
That's barter.
So, barter can't be capitalism. I think you are wrong.
Barter like capitalism is a FREE exchange of goods and services..emphasis on FREE. The only difference being barter does not involve the use of money, where capitalism can.
The essence, really,of capitalism, is the idea that each person is entitled to full use and control of what rightfully belongs to him, and that this includes a right to enter into a deal with someone else, whereby each gives the other some of what is his, in exchange for what he receives from the other. Money doesn't necessarily have to have anything to do with it, and neither does any concept of government.
Supose I'm a caveman, and I've spent my day gathering rocks. I have more rocks than I need, but I'd like to have some sticks.
Og spent his day gathering sticks, so he has more sticks than he needs, but he'd like to have some rocks.
A socialist might look at my supply of rocks, decide that I have more than I need, and that Og has none, and think that some of my rocks should be taken from me, and given to Og. That same socialist might look at Og's supply of sticks, and by the same reasoning, decide that some of Og's sticks should be taken from him and given to me. It's notable that this approach requires an outside authority—a government—to determine what is the fair allocation of sticks and rocks, and to take from those deemed to have too much of something, to be given to those deemed not to have enough.
A capitalist approach would recognize that the rocks that I have gathered belong to me,and that I alone have any right to determine what to do with them, and that the same applies to Og's sticks. It would be entirely up to Og and me to make a deal between us, if I think I need some sticks, and he thinks he needs some rocks, wherein I would give him some of my rocks,and he would give me some of his sticks. No concept of government would necessarily have anything to do with such an exchange. That, is capitalism, in its purest form.