Yes. You should keep it away from oxygen.
Which is a lot harder than you think. Chemists and material scientists spend thousands and thousands of dollars to be able to keep things oxygen free.
Or you could use a (more) stable ascorbic acid derivative like magnesium ascorbyl phosphate (for water phases), ethyl ascorbate or glucoside (amphiphilic), ascorbyl palmitate, tetrahexadecyl ascorbate (oil phase), but all of them come with complications (for example adjusting the pH, or working with enough oil phase to dissolve the ascorbic acid derivative).
If you want to stick with sodium ascorbate or ascorbic acid then you should use a different approach.