yes, you can net the gains against the losses.. if that results in an net loss, you are correct that only $3000 of the loss can be used each year to reduce your other income.
so in your example, you would report a $20,000 net capital gain (long term)
note that you can't use $3,000 of the loss and reduce your ordinary income first and then report a long term capital gain of $23,000 (using your example). it doesn't work that way.
