Our little one has the same, hospitalised several times with skin infections , where skin went from normal to infected in a matter of hours.
Baby is EBF so we didn't have the issue with finding a formula that was allergen free, but I cut out the following allergens from my diet, and now also baby's diet:
Dairy
Egg
Soya (now reintroduced)
Gluten/wheat
Nuts
Sesame
We also changed washing liquid to surcare, no conditioner. Set to the maximum number of rinses for the machine too.
We only bathe baby when really really needs it. The hospital said to bathe daily, but their skin is 10000x worse after bathing.
Baby's skin mostly cleared within 2-3 months (takes up to 8 weeks to get out of everyone's systems).
Baby is weaning now and finding they regularly get triggered but, I'm assuming, unknown allergens or contamination. But this is far more manageable.
In terms of management, we are now off steroids completely and our routine is ..
Morning and night we have three steps:
100% Aloe Vera all over to add moisture
Then
Aproderm all over the top
Then
50/50 to seal the moisture in.
Piriton daily at night to help with itching and sleep. We also found cetirazine worked well but you can't get that for babies in the UK (we got it when abroad where you couldn't get piriton for babies weirdly!).
During the day we just use the Aproderm at every nappy change.
Everything but the aloe is on prescription.
We add dead sea salta when baby does bathe, and add nothing else.
We also used probiotic since birth due to all the antibiotics baby was having to have until we got the eczema under control.
We also had blood allergy tests following the hospital admission (which confirmed ige-allergy to dairy, wheat, eggs and sesame, but didn't test nuts which baby 100% reacts to. Tests said no allergy to soya which we have reintroduced, but there is still potential for a non-ige reactiona to that and other things - which won't show on any allergy test). We also had paediatric dermatology referral (yet to be seen by them, been waiting since January!).
Also note, current advice is don't add anything food related to skin until it is eaten in normal quantities, as it can lead to more allergies.