If you wanted to be even more random - evenly distributed and independent - with dice, you could get a 20-sided dice but then collect the data in base-2 by saying every even number was 1 and odd number was zero,
The thing is, a 20 sided dice has more biases than a 6 sided dice. The more sides the more bias. So that might not be a good idea.
Be very careful about the kind of dice you use because some brands of 6-sided dice are intentionally biased for larger numbers. In any case I suppose that a 2-sided dice (i.e. a coin) has the least bias of them all, and for this particular application has the benefit of being a factor of 256 (exactly 256 tosses).