Sorry to hear about this mrmump and I hope that the many parents who have children with eczema will soon be along with their experiences and advice. Do take time to read through the wealth of posts (also check in the allergy section).
How old is your DD?
Our DD had severe eczema, which got infected on a regular basis. We washed it gently 1-3 times a day (if bathing, make sure you pat dry not rub and moisturise; choice of moisturiser - there are so many, it is a matter of working out which will work best for your DD. Many posters recommend Aveeno but this made our DD's eczema worse) especially before applying the antibiotic ointment. If you have been given an antibiotic cream make sure you use it as prescribed and don't miss out on any doses. We found an ointment better than a cream as it is thicker in consistency and 'sticks' better.
100 % cotton clothing, scratch mitts (prevent scratching although DD may still buff skin), washing powder/detergent used (non-bio, for sensitive skins).
Hydrocortisone - this is useful in helping keep symptoms at bay and also to reduce a flare up but personally, I odn't like to use it all the time. If you can get the eczema under control, or figure out what is setting it off (there isn't always a recognisable cause though), that is the ideal but do not discount how important hydrocortisone is in helping to reduce symptoms and make eczema manageable. Check re. use on broken skin (I think it cannot be used).
Keep a diary with as much detail as you can - what you are doing, where you are going, what is being given to eat, if BF what the mother is eating. It may help you recognise a pattern.
DD has also been effectively trained not to scratch. It meant we spent a lot of time making sure that she didn't and telling her not to.
I know that other posters will come along and there is probably a lot I left out and also points that others have different views about/approaches. The toughest part is that there isn't a single solution but you will work it out although it takes time, patience and perseverance.
Hope DD improves soon.