In the core books, no
There's a reason for that. Getting all of the half-elf's bonuses, see @UrhoKarila's answer, and additional elven bonuses would be far more powerful starting out compared to all other choices. The bonuses found on p. 38-39 of the PHB is a mix/balance between elf and human ancestry. (It's a pretty good deal, all around).
But they can get selected elf race bonuses (for a price) if you use the Sword Coast Adventurer's Guide ...
The Half Elf Variant (listed on p. 116 of the Sword Coast Adventurer's Guide) allows a Half Elf to trade in the two skill bonuses for:
A Half Wood Elf can get Elf Weapon Training(p. 23 PHB, high elf
feature), Fleet of Foot(p. 24 PHB wood elf feature), or Mask of
the Wild(p. 24 PHB, wood elf feature).
A Half Sun or Moon Elf (SCAG p. 106) can get Elf Weapon Training (see
above) or a free Cantrip.
- A Half Drow elf can choose Drow Magic(p. 24 PHB)
A Half Aquatic elf can choose a swimming speed of 30 feet.
This Variant is only available if the DM allows it, just as the Variant Human (with a feat at first level (p.31 PHB)) is only available if the DM allows it.
What is allowed at your table? Ask your DM.
If your table's source material is limited to the Players Handbook, the Monster Manual, and the Dungeon Master's Guide, then the Half Elf bonuses are as stated by @UrhoKarila and in the PHB:
Ability Score Increase of CHA +2; two other ability scores increase
Darkvision to 60'.
Fey Ancestry: advantage on saving throws against being charmed,
magic can’t put you to sleep.
Proficiency in two skills of your choice.
Common, Elvish, and one extra language of your choice.
- Read the Players Handbook material on races;
- Read the Sword Coast Adventurer's Guide material on races (if it's available / allowed,);
- Discuss your options with your DM.