Are you ready for an alternative viewpoint!?!? It's important to consider the context of flu and eczema, because if we define success by reduction of one symptom we may be heading for something bigger.
A couple of questions: did you have a flu jab? This might be a reaction to the jab (which can also cause flu, incidentally). Secondly, how did you treat the flu? Suppression of fever by paracetemol is known to increase eczema and asthma.
First things first, flu jab does not reduce flu or its complications. Despite this there's always a spokesperson willing to say something irrelevant like 'but old people are at risk so we have to vaccinate them', which is illogical if the jab doesn't work.
Secondly, flu is a healing process! Especially fever - it's a physiological process that increases immune activity and purges the body of cancer cells. I look forward to my twice-yearly flu! If it is suppressed, the benefit is less likely, and chronic bedding down of some problems is more likely. Fever is not dangerous (Febrile convulsions in infants is another story, but the risks are massively overstated, haven't got time to go into this here, but crucially, bringing down a high fever doesn't reduce convulsions).
Seasonal flu and colds are as regular and important as leaves falling off trees! When they stop we should worry. In fact colds and flu are sometimes provoked when treating problems such as - eczema! So another bout of flu after christmas would be ideal! If it happens, work with your practitioner to make the most of it, but do not suppress!
But the plot thickens: flu isn't contagious (epidemiological studies show this, look up Hoyle and Wickramasinghe) and it isn't caused by a virus - although virus may sometimes appear as a result, most flu doesn't even involve a virus, so they call it 'flu like illness'. In the only ever attempt in lab conditions to infect people with flu not one subject caught it out of 62 subjects. The only sufferer was the doctor in charge of the experimental facility, and he died! (look up Deer Island Swine Flu experiment)
So the big lesson is in the corect response to flu - call in sick, go to bed and drink plenty of water. It is complications, bronchitis, pneumonia etc that are the real danger; again, another story, but loosely these are a result of exhaustion, which is why absolute rest is so important.
So much depends on how your flu was treated, but...
are the bowels moving? And are you putting anything on the skin? Please don't - but if you are suffering horribly then cold water on it at home, or aloe vera if you're out and about may help. The main thing is get an individual constitutional diagnosis and take it from there.