If I mix carbon dioxide and sodium hydroxide in the right amounts in water, do I get baking soda?
This is certainly one method by which sodium bicarbonate can be produced.  According to this Wikipedia article:

$\ce{NaHCO3}$ may be obtained by the reaction of carbon dioxide with an aqueous solution of sodium hydroxide. The initial reaction produces sodium carbonate:
$$\ce{CO2 + 2 NaOH → Na2CO3 + H2O}$$
Further addition of carbon dioxide produces sodium bicarbonate, which at sufficiently high concentration will precipitate out of solution:
$$\ce{Na2CO3 + CO2 + H2O → 2 NaHCO3}$$

The net reaction for this two-step process is then simply:
$$\ce{CO2 + NaOH -&gt; NaHCO3}$$
Note that this is not the common method by which sodium bicarbonate is produced commercially, but it is a perfectly valid method of it's synthesis.