Can I store all channels from the signal coming from a digital antenna?
You can, and this is what a software defined radio does. The only requirement is that your sample rate in the recording is sufficiently high to capture the entire range of frequencies you wish to record, and that the bits per sample provide sufficient dynamic range.

It should be noted that a "digital antenna" actually outputs an analog signal. In fact it's not digital in any sense, except that it's designed to receive the frequencies used for digital television. The signal does not become digital until after it's demodulated by your TV.

Depending on regional details of course, the digital TV broadcast band goes from approximately 470 to 700 MHz. That means the band is 700 - 470 = 230 MHz wide, meaning a sample rate at least twice that is required: 460 MHz. Considering the typical computer sound card (and MP3s, CDs, etc) typically has a sample rate of 0.048 MHz, recording the entire TV broadcast band would require some specialized equipment, and the recording would be relatively large.