As you've noticed, the "u" sound in Japanese is often reduced when it comes between voiceless consonants. Since "s" and "k" are both voiceless, Japanese people say "skiyaki" and "asaksa".
I think this confusion is also a result of the fact that the "u" sound in Japanese is different from that in English. Since we're not familiar with this sound, we hear something between "skiyaki" and "sookiyaki" even when Japanese people say "sukiyaki" clearly.
The difference between the "oo" sound that we hear and the "u" sound that Japanese people produce is like the difference between the vowel sounds in "boot" and "foot". In "boot," our lips are rounded and protruding. In "foot" our lips are neither rounded nor protruding (at least not much). That's like the Japanese "u" sound.(Interestingly, if you say "foot" quickly, you'll notice it also becomes "ft".)
Looking at IPA charts, I see there are some differences between the vowel sound in foot and the Japanese "u" sound, but I'm pretty sure that if you pronounce the "u" sound in "sukiyaki" like the one in "foot," it will be indistinguishable to a native speaker.
I've just started my foray into phonology, so excuse me if I get my terms wrong!